Question 1: Palette 1/138

AP-EAPCET 2025 MCQ

If $\frac{x^4}{(x-1)(x-2)}=f(x)+\frac{A}{x-1}+\frac{B}{x-2}$, then $f(-2)+A+B=$

A

32

B

28

C

22

D

20

Question 2: Palette 2/138

AP-EAPCET 2025 MCQ

If $\int \frac{x^4+1}{x^2+1} d x=A x^3+B x^2+C x+D \tan ^{-1} x+E$, then $A+B+C+D=$

A

$\frac{3}{2}$

B

$\frac{4}{3}$

C

$\frac{1}{3}$

D

$\frac{2}{3}$

Question 3: Palette 3/138

AP-EAPCET 2025 MCQ

$ \begin{aligned} & \text { If } \int \frac{x^2-x+2}{x^2+x+2} d x=x-\log (f(x))+\frac{2}{\sqrt{7}} \tan ^{-1}(g(x))+c, \text { then } \\ & f(-1)+\sqrt{7} g(-1)= \end{aligned} $

A

1

B

0

C

-1

D

2

Question 4: Palette 4/138

AP-EAPCET 2025 MCQ

$ \int \sec \left(x-\frac{\pi}{3}\right) \sec \left(x+\frac{\pi}{6}\right) d x= $

A

$\log \left|\frac{\sec \left(x-\frac{\pi}{3}\right)}{\sec \left[x+\frac{\pi}{6}\right]}\right|+C$

B

$\log \left|\frac{\cos \left(x-\frac{\pi}{3}\right)}{\cos \left(x+\frac{\pi}{6}\right)}\right|+C$

C

$\log \left|\frac{\operatorname{cosec}\left(x-\frac{\pi}{3}\right)}{\operatorname{cosec}\left(x+\frac{\pi}{6}\right)}\right|+C$

D

$\log \left|\frac{\sin \left(x-\frac{\pi}{3}\right)}{\sin \left(x+\frac{\pi}{6}\right)}\right|+C$

Question 5: Palette 5/138

AP-EAPCET 2025 MCQ

If $\int \frac{a \cos x+3 \sin x}{5 \cos x+2 \sin x} d x=\frac{26}{29} x-\frac{k}{29} \log |5 \cos x+2 \sin x|+\ldots$ then $|a+k|=$

A

3

B

11

C

12

D

2

Question 6: Palette 6/138

AP-EAPCET 2025 MCQ

If $\int \frac{d x}{1-\sin ^4 x}=A \tan x+B \tan ^{-1}(\sqrt{2} \tan x)+C$, then $A^2-B^2=$

A

$\frac{1}{2}$

B

$\frac{3}{4}$

C

$\frac{1}{4}$

D

$\frac{1}{8}$

Question 7: Palette 7/138

AP-EAPCET 2025 MCQ

If $\frac{x^2}{\left(x^2+2\right)\left(x^4-1\right)}=\frac{A}{x^2-1}+\frac{B}{x^2+1}+\frac{C}{x^2+2}$, then $A+B-C=$

A

0

B

$\frac{4}{3}$

C

$\frac{3}{4}$

D

2

Question 8: Palette 8/138

AP-EAPCET 2025 MCQ

If $\int \frac{5 \tan x}{\tan x-2} d x=a x+b \log |\sin x-2 \cos x|+c$, then $a+b=$

A

2

B

3

C

4

D

-1

Question 9: Palette 9/138

AP-EAPCET 2025 MCQ

$ \int x \cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right) d x(x>0)= $

A

$-x+\left(1+x^2\right) \tan ^{-1} x+C$

B

$x-\left(1+x^2\right) \cot ^{-1} x+C$

C

$-x+\left(1+x^2\right) \cot ^{-1} x+C$

D

$x-\left(1+x^2\right) \tan ^{-1} x+C$

Question 10: Palette 10/138

AP-EAPCET 2025 MCQ

$ \int \frac{d x}{(1+\sqrt{x}) \sqrt{x-x^2}}= $

A

$-2 \sqrt{\frac{1+\sqrt{x}}{1-\sqrt{x}}}+C$

B

$-\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}+C$

C

$-2 \sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}+C$

D

$2 \sqrt{\frac{1+\sqrt{x}}{1-\sqrt{x}}}+C$

Question 11: Palette 11/138

AP-EAPCET 2025 MCQ

$ \int \sin ^{-1}\left(\sqrt{\frac{x}{a+x}}\right) d x= $

A

$(a+x) \tan ^{-1} \sqrt{\frac{x}{a}}+a x+C$

B

$(a+x) \tan ^{-1} \sqrt{\frac{x}{a}}+\sqrt{a x}+C$

C

$(a+x) \tan ^{-1} \sqrt{\frac{a}{x}}-\sqrt{a x}+C$

D

$(a+x) \tan ^{-1} \sqrt{\frac{x}{a}}-\sqrt{a x}+C$

Question 12: Palette 12/138

AP-EAPCET 2025 MCQ

If $\int \frac{x}{x \tan x+1} d x=\log f(x)+k$, then $f\left(\frac{\pi}{4}\right)=$

A

$\frac{\pi}{4 \sqrt{2}}$

B

$\pi+\frac{\pi}{2 \sqrt{2}}$

C

$\frac{\pi+4}{4 \sqrt{2}}$

D

$\frac{\pi-4}{4 \sqrt{2}}$

Question 13: Palette 13/138

AP-EAPCET 2025 MCQ

If $\frac{2 x^4-3 x^2+4}{\left(x^2+1\right)\left(x^2+2\right)}=a+\frac{p x+q}{x^2+1}+\frac{m x+n}{x^2+2}$, then $\frac{n}{q}=$

A

$p+m-a$

B

$\frac{p+m}{a}$

C

$\frac{a}{p+m}$

D

$p+m+a$

Question 14: Palette 14/138

AP-EAPCET 2025 MCQ

$ \int(\log 2 x)^3 d x= $

A

$x\left((\log 2 x)^3-3(\log 2 x)^2+6(\log 2 x)-6\right]+C$

B

$\frac{x}{4}\left[4(\log 2 x)^3-6(\log 2 x)^2+6(\log 2 x)-3\right]+C$

C

$\frac{x}{2}\left[(\log 2 x)^3-3(\log 2 x)^2+3(\log 2 x)-6\right]+C$

D

$x\left[(\log 2 x)^3-6(\log 2 x)^2+18(\log 2 x)-54\right]+C$

Question 15: Palette 15/138

AP-EAPCET 2025 MCQ

$ \int \frac{x+1}{(x-2) \sqrt{1-x}} d x= $

A

$\log (x+1)-\log (x-2) \sqrt{1-x}+C$

B

$\log (x-2) \sqrt{1-x}+C$

C

$6 \tan ^{-1} \sqrt{1-x}-2 \sqrt{1-x}+C$

D

$4 \tan ^{-1} \sqrt{1-x}-2 \sqrt{1-x}+C$

Question 16: Palette 16/138

AP-EAPCET 2025 MCQ

$ \int \frac{1}{1+x+x^2} d x= $

A

$\frac{2}{\sqrt{3}} \log \left(\frac{2 x+1+\sqrt{3}}{2 x-1-\sqrt{3}}\right)+C$

B

$\frac{1}{\sqrt{3}} \log \left(\frac{2 x+1-\sqrt{3}}{2 x+1+\sqrt{3}}\right)+C$

C

$\frac{2}{\sqrt{3}} \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)+C$

D

$\frac{2}{\sqrt{5}} \tan ^{-1}\left(\frac{2 x+1}{\sqrt{5}}\right)+C$

Question 17: Palette 17/138

AP-EAPCET 2025 MCQ

If $\int \frac{d x}{(x \tan x+1)^2}=f(x)+C$, then $\lim\limits_{x \rightarrow \frac{\pi}{2}} f(x)=$

A

$\frac{\pi}{2}$

B

$\frac{2}{\pi}$

C

$\frac{1}{\pi}$

D

$\infty$

Question 18: Palette 18/138

AP-EAPCET 2025 MCQ

$ \int \sin ^3 x \cos ^2 x d x= $

A

$\frac{\sin ^4 x \cos x}{5}-\frac{\sin ^2 x \cos x}{15}-\frac{2 \cos x}{15}+C$

B

$-\frac{\sin ^4 x \cos x}{5}-\frac{\sin ^2 x \cos x}{15}+\frac{2 \cos x}{15}+C$

C

$\frac{\sin ^4 x \cos ^{\prime} x}{5}-\frac{\sin ^2 x \cos x}{15}+\frac{2 x}{15}+C$

D

$\frac{\sin ^4 x \cos x}{5}+\frac{\sin ^2 x \cos x}{3}-\frac{2 x}{15}+C$

Question 19: Palette 19/138

AP-EAPCET 2025 MCQ

19. If $\frac{3 x^3-7 x+1}{(x-2)^5}=\frac{A}{x-2}+\frac{B}{(x-2)^2}+\frac{C}{(x-2)^3}+\frac{D}{(x-2)^4}+\frac{E}{(x-2)^5}, \text { then } A(B+C+D+E)= $

A

0

B

348

C

64

D

256

Question 20: Palette 20/138

AP-EAPCET 2025 MCQ

$ \int(\sqrt{\tan x}+\sqrt{\cot x}) d x= $

A

$2 \tan ^{-1}\left(\frac{\tan x-1}{\sqrt{\tan x}}\right)+C$

B

$\tan ^{-1}\left(\frac{\tan x-2}{2 \sqrt{\tan x}}\right)+C$

C

$\sqrt{2} \tan ^{-1}\left(\frac{\tan x-1}{\sqrt{2 \tan x}}\right)+C$

D

$\sqrt{2} \tan ^{-1}\left(\frac{\tan x+1}{\sqrt{2} \tan x}\right)+C$

Question 21: Palette 21/138

AP-EAPCET 2025 MCQ

$\int \frac{\sqrt{x-2}}{2 x+4} d x=$

A

$\sqrt{x-2}-\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{x-2}}{2}\right)+C$

B

$\sqrt{x-2}-2 \tan ^{-1}\left(\frac{\sqrt{x-2}}{2}\right)+C$

C

$\sqrt{x-2}+2 \tan ^{-1}\left(\frac{\sqrt{x-2}}{2}\right)+C$

D

$\sqrt{x-2}+\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{x-2}}{2}\right)+C$

Question 22: Palette 22/138

AP-EAPCET 2025 MCQ

If $\int x^{49}\left[\tan ^{-1} x^{50}+\frac{x^{50}}{1+x^{100}}\right] d x=\frac{x^n}{k} f(x)+c$, then

$ f(x)-f\left(\sqrt[k]{x^n}\right)= $

A

$k+n$

B

$k-n$

C

$1 / k$

D

$1 / n$

Question 23: Palette 23/138

AP-EAPCET 2025 MCQ

$ \int \frac{x}{\sqrt{x^2-2 x+5}} d x= $

A

$\sqrt{x^2-2 x+5}+\sinh ^{-1}\left(\frac{x-1}{2}\right)+C$

B

$\frac{1}{2} \sqrt{x^2-2 x+5}+\sin ^{-1}\left(\frac{x-1}{2}\right)+C$

C

$2 \sqrt{x^2-2 x+5}+\cosh ^{-1}\left(\frac{x-1}{2}\right)+C$

D

$\sqrt{x^2-2 x+5}-\cos ^{-1}\left(\frac{x-1}{2}\right)+C$

Question 24: Palette 24/138

AP-EAPCET 2025 MCQ

For $0 < x < 1, \int\left[\tan ^{-1}\left(1-x+x^2\right)+\tan ^{-1}(1-x)\right] d x=$

A

$x \cot ^{-1} x+\log \sqrt{1+x^2}+C$

B

$x \tan ^{-1} x-\log \left(1+x^2\right)+C$

C

$x \cot ^{-1} x+\frac{3}{4} \log \left(1+x^2\right)+C$

D

$x \tan ^{-1} x-\frac{3}{4} \log \sqrt{1+x^2}+C$

Question 25: Palette 25/138

AP-EAPCET 2025 MCQ

If $\frac{3 x+1}{(x-1)\left(x^2+2\right)}=\frac{A}{x-1}+\frac{B x+C}{x^2+2}$, then $5(A-B)=$

A

$A+C$

B

8 C

C

$C+8$

D

$\frac{C}{8}$

Question 26: Palette 26/138

AP-EAPCET 2025 MCQ

$ \int \frac{\sec ^2 x}{\sin ^7 x} d x-\int \frac{7}{\sin ^7 x} d x= $

A

$\frac{1}{\sin ^6 x \cos x}+C$

B

$\frac{\tan x}{\sin ^8 x}+C$

C

$\sin ^8 x \cos x+C$

D

$\sec x \tan ^7 x+C$

Question 27: Palette 27/138

AP-EAPCET 2025 MCQ

If $\int\left(x^6+x^4+x^2\right) \sqrt{2 x^4+3 x^2+6} d x=f(x)+c$, then $f(3)=$

A

$\frac{3}{2}(95)^{3 / 2}$

B

$\frac{3}{2}(195)^{3 / 2}$

C

$\frac{3}{2}(265)^{3 / 2}$

D

$\frac{3}{2}(175)^{3 / 2}$

Question 28: Palette 28/138

AP-EAPCET 2025 MCQ

$ \int \frac{d x}{(x+1) \sqrt{x^2+1}}= $

A

$\frac{1}{\sqrt{2}} \sinh ^{-1}\left(\frac{1+x}{1-x}\right)+C$

B

$\frac{1}{\sqrt{2}} \sinh ^{-1}\left(\frac{1-x}{1+x}\right)+C$

C

$-\frac{1}{\sqrt{2}} \sinh ^{-1}\left(\frac{1-x}{1+x}\right)+C$

D

$-\frac{1}{\sqrt{2}} \sinh ^{-1}\left(\frac{1+x}{1-x}\right)+C$

Question 29: Palette 29/138

AP-EAPCET 2025 MCQ

If $\int \frac{d x}{2 \cos x+3 \sin x+4}=\frac{2}{\sqrt{3}} f(x)+C$, then $f\left(\frac{2 \pi}{3}\right)=$

A

$\frac{\pi}{12}$

B

$\frac{\pi}{8}$

C

$\frac{5 \pi}{12}$

D

$\frac{5 \pi}{8}$

Question 30: Palette 30/138

AP-EAPCET 2025 MCQ

If $\int \frac{1}{\left((x+4)^3(x+1)^5\right)^{1 / 4}} d x=A \cdot\left(\frac{x+4}{x+1}\right)^n+C$

A

$n$ A=3

B

$n+\frac{1}{A}=-\frac{1}{2}$

C

$A+n=1$

D

$A=n$

Question 31: Palette 31/138

AP-EAPCET 2025 MCQ

$ \int \frac{x+1}{x^3-1} d x= $

A

$\frac{1}{3} \log \left(\frac{x+1}{x^2+x+1}\right)+C$

B

$\frac{1}{3} \log \left(\frac{(x-1)^2}{x^2+x+1}\right)+C$

C

$\frac{1}{3} \log \left(\frac{x-1}{x^2+x+1}\right)+C$

D

$\frac{1}{3} \log \left(\frac{(x+1)^2}{x^2-x+1}\right)+C$

Question 32: Palette 32/138

AP-EAPCET 2025 MCQ

$ \int \frac{x^4-16 x^2+2 x+8}{x^3-4 x^2+2} d x= $

A

$\frac{x^2+8 x+C}{2}$

B

$x^2+8 x+C$

C

$x^3-4 x+C$

D

$\frac{x^2-8 x+C}{2}$

Question 33: Palette 33/138

AP-EAPCET 2025 MCQ

$ \int \frac{\sec ^2 x}{(\sec x+\tan x)^{\frac{5}{2}}} d x= $

A

$-\frac{(\sec x+\tan x)^{\frac{5}{2}}}{5}-\frac{(\sec x+\tan x)^{\frac{7}{2}}}{7}+C$

B

$-\frac{(\sec x-\tan x)^{\frac{5}{2}}}{5}-\frac{(\sec x-\tan x)^{\frac{7}{2}}}{7}+C$

C

$-\frac{(\sec x+\tan x)^{\frac{3}{2}}}{3}-\frac{(\sec x+\tan x)^{\frac{7}{2}}}{7}+C$

D

$-\frac{(\sec x-\tan x)^{\frac{3}{2}}}{3}-\frac{(\sec x-\tan x)^{\frac{7}{2}}}{7}+C$

Question 34: Palette 34/138

AP-EAPCET 2025 MCQ

$ \int \frac{1}{\cos x}\left[\frac{1}{\sin x}-\frac{1}{\sin x+3 \cos x}\right] d x= $

A

$\frac{1}{3} \log \left|\frac{\sin x}{\sin x+3 \cos x}\right|+C$

B

$\log \left|\frac{\cos x}{\sin x+3 \cos x}\right|+c$

C

$\frac{1}{3} \log \left|\frac{\cos x}{\sin x+3 \cos x}\right|+C$

D

$\log \left|\frac{\sin x}{\sin x+3 \cos x}\right|+c$

Question 35: Palette 35/138

AP-EAPCET 2025 MCQ

$ \int \cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right) d x= $

A

$2\left[x \tan ^{-1} x-\log \sqrt{1+x^2}\right]+C$

B

$2 x \tan ^{-1} x+\log \sqrt{1+x^2}+C$

C

$x \tan ^{-1} x+\log \sqrt{1-x^2}+C$

D

$2\left[\tan ^{-1} x-\log \sqrt{1+x^2}\right]+C$

Question 36: Palette 36/138

AP-EAPCET 2025 MCQ

If $\frac{a x+5}{\left(x^2+b\right)(x+3)}=\frac{x+21}{12\left(x^2+b\right)}+\frac{c}{12(x+3)}$, then $b^2=$

A

$a^3-c$

B

$a^2+c$

C

$a-c$

D

$a+c$

Question 37: Palette 37/138

AP-EAPCET 2025 MCQ

$ \int\left(\sum_{r=0}^{\infty} \frac{x^r 2^r}{r!}\right) d x= $

A

$e^x+C$

B

$\frac{-2}{1-2 x}+C$

C

$2 e^{2 x}+C$

D

$\frac{e^{2 x}}{2}+C$

Question 38: Palette 38/138

AP-EAPCET 2025 MCQ

$ \int \frac{d x}{12 \cos x+5 \sin x}= $

A

$\frac{1}{13} \log \left|\tan \left(\frac{\pi}{4}+\frac{x}{2}-\frac{1}{2} \tan ^{-1} \frac{5}{12}\right)\right|+C$

B

$\frac{5}{12} \log \left|\tan \left(\frac{\pi}{4}+\frac{x}{2}-\frac{1}{2} \tan ^{-1} \frac{5}{12}\right)\right|+C$

C

$\frac{1}{13} \log \left|\tan \left(\frac{\pi}{4}+\frac{x}{2}+\frac{1}{2} \tan ^{-1} \frac{5}{12}\right)\right|+C$

D

$\frac{5}{12} \log \left|\tan \left(\frac{\pi}{4}+\frac{x}{2}+\frac{1}{2} \tan ^{-1} \frac{5}{12}\right)\right|+C$

Question 39: Palette 39/138

AP-EAPCET 2025 MCQ

If $\int \frac{\cos ^3 x}{\sin ^2 x+\sin ^4 x} d x=c-\operatorname{cosec} x-f(x)$, then $f\left(\frac{\pi}{2}\right)=$

A

1

B

0

C

$\pi / 2$

D

$\pi$

Question 40: Palette 40/138

AP-EAPCET 2025 MCQ

$ \int \frac{13 \cos 2 x-9 \sin 2 x}{3 \cos 2 x-4 \sin 2 x} d x= $

A

$3 x-\frac{1}{2} \log |3 \cos 2 x-4 \sin 2 x|+C$

B

$\frac{x}{2}-3 \log |3 \cos 2 x-4 \sin 2 x|+C$

C

$3 x+\frac{1}{2} \log |3 \cos 2 x-4 \sin 2 x|+C$

D

$x+\frac{3}{2} \log |3 \cos 2 x-4 \sin 2 x|+C$

Question 41: Palette 41/138

AP-EAPCET 2025 MCQ

$ \int \sqrt{x^2+x+1} d x $

A

$\frac{(2 x+1)}{4} \sqrt{x^2+x+1}+\frac{3}{8} \sinh ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)+C$

B

$\frac{x+1}{4} \sqrt{x^2+x+1}+\frac{3}{8} \sinh ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)+C$

C

$\frac{x+1}{4} \sqrt{x^2+x+1}-\frac{3}{8} \sinh ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)+C$

D

$\frac{(2 x+1)}{4} \sqrt{x^2+x+1}-\frac{3}{8} \sinh ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)+C$

Question 42: Palette 42/138

AP-EAPCET 2025 MCQ

If $\frac{x+1}{(x-1)^2\left(x^2+1\right)}=\frac{A}{x-1}+\frac{B}{(x-1)^2}+\frac{C x+D}{x^2+1}$, then $\sqrt{3 A^2+4 D^2+5 C^2+B^2}=$

A

$\frac{3}{2}$

B

$\frac{1}{2}$

C

1

D

2

Question 43: Palette 43/138

AP-EAPCET 2025 MCQ

$ \int \frac{1}{9 \cos ^2 x-24 \sin x \cos x+16 \sin ^2 x} d x= $

A

$\frac{\cos x}{4(3 \cos x-4 \sin x)}+C$

B

$\frac{\sin x}{4(3 \cos x-4 \sin x)}+C$

C

$\frac{\cos x}{3 \cos x-4 \sin x}+C$

D

$\frac{\sin x}{3 \cos x-4 \sin x}+C$

Question 44: Palette 44/138

AP-EAPCET 2025 MCQ

If $\int \frac{1}{\cot \frac{x}{2} \cot \frac{x}{3} \cot \frac{x}{6}} d x=A \log \left|\cos \frac{x}{2}\right| +B \log \left|\cos \frac{x}{3}\right|+C \log \left|\cos \frac{x}{6}\right|+k$, then $A+B+C=$

A

7

B

-7

C

11

D

1

Question 45: Palette 45/138

AP-EAPCET 2025 MCQ

$ \int \frac{\sin x+\cos x}{\sin x-\cos x} d x= $

A

$-x+\log |\cos x-\sin x|+C$

B

$x-\log |\cos x-\sin x|+C$

C

$-\log |\cos x-\sin x|+C$

D

$\log |\cos x-\sin x|+C$

Question 46: Palette 46/138

AP-EAPCET 2025 MCQ

$ \int \frac{x^4-1}{x^2 \sqrt{x^4+x^2+1}} d x= $

A

$\frac{2 \sqrt{x^4+x^2+1}}{x}+C$

B

$\frac{\sqrt{x^4+x^2+1}}{x}+C$

C

$\frac{\sqrt{x^4+x^2+1}}{2 x}+C$

D

$\frac{4 \sqrt{x^4+x^2+1}}{x}+C$

Question 47: Palette 47/138

AP-EAPCET 2025 MCQ

$ \int \frac{(3 x-2) \tan \left(\sqrt{9 x^2-12 x+1}\right)}{\sqrt{9 x^2-12 x+1}} d x= $

A

$\frac{1}{3} \sec ^2 \sqrt{9 x^2-12 x+1+C}$

B

$\frac{1}{3} \sec ^2 x+C$

C

$\frac{1}{2} \log \left|\sec \sqrt{9 x^2-12 x+1}\right|+C$

D

$\frac{1}{3} \log \left|\sec \sqrt{9 x^2-12 x+1}\right|+C$

Question 48: Palette 48/138

AP-EAPCET 2025 MCQ

If $\int e^{\sin x}(1+\sec x \tan x) d x=e^{\sin x} f(x)+c$, then in $0 \leq x \leq 2 \pi$, then number of solutions of $f(x)=1$ is

A

0

B

4

C

3

D

2

Question 49: Palette 49/138

AP-EAPCET 2025 MCQ

If $\int \frac{d x}{(x-1)^{\frac{3}{2}}(x-3)^{\frac{1}{2}}}=\sqrt{f(x)}+C$, then $f(-1)-f(0)=$

A

-3

B

-4

C

-2

D

-1

Question 50: Palette 50/138

AP-EAPCET 2025 MCQ

$ \int \frac{x}{\left(1-x^2\right) \sqrt{2-x^2}} d x= $

A

$\log \left|\frac{\sqrt{2-x^2}+1}{\sqrt{2-x^2}-1}\right|+C$

B

$\frac{1}{2} \log \left|\frac{\sqrt{2-x^2}}{1-x^2}\right|+C$

C

$\frac{1}{2} \log \left|\frac{1+\sqrt{2-x^2}}{1-\sqrt{2-x^2}}\right|+C$

D

$\log \left|\frac{1-x^2}{\sqrt{2-x^2}}\right|+C$

Question 51: Palette 51/138

AP-EAPCET 2025 MCQ

$\int\left(\frac{1+x+\sqrt{x+x^2}}{\sqrt{x}+\sqrt{1+x}}\right) d x=$

A

$\frac{1}{2} \sqrt{1+x}+C$

B

$\frac{2}{3}(1+x)^{\frac{3}{2}}+C$

C

$\sqrt{1+x}+C$

D

$2(1+x)^{\frac{3}{2}}+C$

Question 52: Palette 52/138

AP-EAPCET 2025 MCQ

If $\int x^2 \cos ^2 x d x=\frac{1}{6} f(x)+g(x) \sin 2 x +h(x) \cos 2 x+c$, then $f(1)+g(2)+h\left(\frac{1}{2}\right)=$

A

0

B

2

C

1

D

-1

Question 53: Palette 53/138

AP-EAPCET 2025 MCQ

$ \int \frac{e^{\sin x}(\sin 2 x-8 \cos x)}{2(\sin x-3)^2} d x= $

A

$e^{\sin x}(\sin x-3)+C$

B

$\frac{e^{\sin x}}{(\sin x-3)^2}+C$

C

$e^{\sin x}(\sin x-3)^2+C$

D

$\frac{e^{\sin x}}{\sin x-3}+C$

Question 54: Palette 54/138

AP-EAPCET 2025 MCQ

If $\int\left(3 t^2 \sin \frac{1}{t}-t \cos \frac{1}{t}\right) d t=f(t) \sin \left(\frac{1}{t}\right)+C$ then $f(2)=$

A

2

B

-12

C

8

D

-16

Question 55: Palette 55/138

AP-EAPCET 2025 MCQ

$ \int(\log x)^3 x^4 d x= $

A

$x^5\left[\frac{1}{5}(\log x)^3-\frac{3}{25}(\log x)^2+\frac{6}{125} \log x-\frac{6}{625}\right]+C$

B

$x^5\left[\frac{1}{5}(\log x)^3-\frac{2}{25}(\log x)^2+\frac{6}{125} \log x-\frac{12}{125}\right]+C$

C

$x^5\left[\frac{1}{5}(\log x)^3-\frac{4}{25}(\log x)^2-\frac{9}{125} \log x-\frac{8}{125}\right]+C$

D

$x^5\left[\frac{1}{5}(\log x)^3+\frac{3}{25}(\log x)^2-\frac{6}{125} \log x-\frac{6}{125}\right]+C$

Question 56: Palette 56/138

AP-EAPCET 2025 MCQ

$ \int \frac{\sin 2 x}{\sin ^2 x+3 \cos x-3} d x $

A

$2 \log \left|\frac{\cos x-2}{\cos x-1}\right|+C$

B

$\log \left(\frac{(\cos x-2)^2}{(\cos x-1)^4}\right)+C$

C

$\log \left(\frac{(\cos x-2)^2}{|\cos x-1|}\right)+C$

D

$\log \left(\frac{(\cos x-2)^4}{(\cos x-1)^2}\right)+C$

Question 57: Palette 57/138

AP-EAPCET 2025 MCQ

If $\int \frac{d x}{\sin ^3 x+\cos ^3 x}=A \log \left|\frac{\sqrt{2}+t}{\sqrt{2}-t}\right|+B \tan ^{-1}(t)+C$, then $\left(\frac{B}{A}, t\right)=$

A

$(3 \sqrt{2}, \sin x-\cos x)$

B

$(2 \sqrt{2}, \sin x-\cos x)$

C

$\left(\frac{\sqrt{2}}{3}, \sin x-\cos x\right)$

D

$\left(\frac{3}{\sqrt{2}}, \sin x+\cos x\right)$

Question 58: Palette 58/138

AP-EAPCET 2024 MCQ

$\frac{4 x^2+5}{(x-2)^4}=\frac{A}{(x-2)}+\frac{B}{(x-2)^2}+\frac{C}{(x-2)^3}+\frac{D}{(x-2)^4}$, then $\sqrt{\frac{A}{C}+\frac{B}{C}+\frac{D}{C}}$ is equal to

A

$\frac{\sqrt{29}}{4}$

B

$\frac{\sqrt{23}}{4}$

C

$\frac{5}{4}$

D

$\frac{4}{5}$

Question 59: Palette 59/138

AP-EAPCET 2024 MCQ

If $\int \frac{\sqrt[4]{x}}{\sqrt{x}+\sqrt[4]{x}} d x=$ $\frac{2}{3}\left[A \sqrt[4]{x^3}+B \sqrt[4]{x^2}+C \sqrt[4]{x}+D \log (1+\sqrt[4]{x})\right]+K$, then $\frac{2}{3}(A+B+C+D)$ is equal to

A

$2 / 3$

B

$-2 / 3$

C

$4 / 3$

D

$-4 / 3$

Question 60: Palette 60/138

AP-EAPCET 2024 MCQ

$\int(\log x)^m x^n d x$ is equal to

A

$\int t^m e^{n t} d t, \quad t=e^x$

B

$\int t^m e^{(n+1) t} d t, \quad t=e^x$

C

$\int t^m e^{(n+1) t} d t, \quad x=e^t$

D

$\int t^m e^{n t} d t, \quad x=e^t$

Question 61: Palette 61/138

AP-EAPCET 2024 MCQ

$\int \sin ^{-1}\left(\sqrt{\frac{x-a}{x}}\right) d x$ is equal to

A

$x \cos ^{-1} \sqrt{\frac{a}{x}}-\sqrt{a x-a^2}+c$

B

$x \sec ^{-1} \cdot \sqrt{\frac{a}{x}}+\sqrt{x^2-a x}+c$

C

$x \sin ^{-1} \sqrt{\frac{x}{a}}+\sqrt{x^2+a x}+c$

D

$\frac{x}{a} \sin ^{-1} \frac{x}{a}+\frac{x^2}{a} \sqrt{1+a^2}+c$

Question 62: Palette 62/138

AP-EAPCET 2024 MCQ

If $\int \frac{\sin x \cos x}{\sqrt{\cos ^4 x-\sin ^4 x}} d x=-\frac{f(x)}{2}+c$, then domain of $f(x)$ is

A

$[2 n \pi,(2 n+1) \pi], n=0,1,2 \ldots$

B

$\left[(4 n-1) \frac{\pi}{2} \cdot(4 n+1) \frac{\pi}{2}\right], n=0,1,2 \ldots$

C

$\left[(4 n-1) \frac{\pi}{4} \cdot(4 n+1) \frac{\pi}{4}\right], n=0,1,2 \ldots$

D

$\left[\left(2 n \frac{\pi}{4} 1 .(2 n+1) \frac{\pi}{4}\right], n=0,1,2 \ldots\right.$

Question 63: Palette 63/138

AP-EAPCET 2024 MCQ

$ \text { If } \frac{13 x+43}{2 x^2+17 x+30}=\frac{A}{2 x+5}+\frac{B}{x+6} \text {, then } A+B \text { is equal to } $

A

8

B

18

C

3

D

5

Question 64: Palette 64/138

AP-EAPCET 2024 MCQ

$\int e^{4 x^2+8 x-4}(x+1) \cos \left(3 x^2+6 x-4\right) d x$ is equal to

A

$\frac{e^{4 x^2+8 x-4}}{25}\left[3 \sin \left(3 x^2+6 x-4\right)-4 \cos \left(3 x^2+6 x-4\right)\right]+c$

B

$\frac{e^{4 x^2+8 x-4}}{50}\left[4 \cos \left(3 x^2+6 x-4\right)+3 \sin \left(3 x^2+6 x-4\right]+c\right.$

C

$\frac{e^{4 x^2+8 x-4}}{25}\left[3 \cos \left(3 x^2+6 x-4\right)+4 \sin \left(3 x^2+6 x-4\right]+c\right.$

D

$\frac{e^{4 x^2+8 x-4}}{50}\left[4 \sin \left(3 x^2+6 x-4\right)+3 \cos \left(3 x^2+6 x-4\right]+c\right.$

Question 65: Palette 65/138

AP-EAPCET 2024 MCQ

$\int\left[(\log 2 x)^2+2 \log 2 x\right] d x$ is equal to

A

$(\log 2 x)^2+c$

B

$2 x \log 2 x+c$

C

$x(\log 2 x)^2+c$

D

$2 x(\log x)^2+c$

Question 66: Palette 66/138

AP-EAPCET 2024 MCQ

If $\int \log \left(6 \sin ^2 x+17 \sin x+12\right) \cos x d x=f(x)+c$, then $f\left(\frac{\pi}{2}\right)$ is equal to

A

$\frac{1}{6}\left[\log 5^5+\log 7^7-12\right]$

B

$\frac{1}{6}[7 \log 5+5 \log 7+29]$

C

$\frac{1}{6}[14 \log 5+15 \log 7+12]$

D

$\frac{1}{6}[15 \log 5+14 \log 7-29]$ $ $

Question 67: Palette 67/138

AP-EAPCET 2024 MCQ

$\int \frac{1}{\left(1+x^2\right) \sqrt{x^2+2}} d x$ is equal to

A

$-\tan ^{-1} \frac{\sqrt{x^2+2}}{|x|}+c$

B

$-\tan ^{-1} \sqrt{x^2+2}+c$

C

$-\tan ^{-1} \sqrt{\frac{x^2+1}{x^2+2}}+c$

D

$-\tan ^{-1} \sqrt{\frac{x^2+2}{x^2+1}}+c$

Question 68: Palette 68/138

AP-EAPCET 2024 MCQ

$\int \sin ^4 x \cos ^4 x d x$ is equal to

A

$\frac{1}{128}\left(-2 \sin ^3 x \cos x-3 \sin x \cos x+3\right)+c$

B

$\frac{1}{256}\left(-2 \sin ^3 2 x \cos 2 x-3 \sin 2 x \cos 2 x+6 x\right)+c$

C

$\frac{1}{128}\left(2 \sin ^3 x \cos x-3 \sin x \cos x+3 x\right)+c$

D

$\frac{1}{256}\left(3 \sin ^3 x \cos x-2 \sin x \cos x+2\right)+c$

Question 69: Palette 69/138

AP-EAPCET 2024 MCQ

$ \int \frac{x^2-1}{x^3 \sqrt{2 x^4-2 x^2+1}} d x $

A

$\frac{1+2 x^2+2 x^4}{2 x^2}+c$

B

$\frac{\left(1+2 x^2+2 x^4\right)^{\frac{1}{2}}}{2 x^2}+c$

C

$\frac{1-2 x^2+2 x^4}{2 x^2}+c$

D

$\frac{\left(1-2 x^2+2 x^4\right)^{\frac{1}{2}}}{2 x^2}+c$

Question 70: Palette 70/138

AP-EAPCET 2024 MCQ

$ \int \frac{x^3 \tan ^{-1} x^4}{1+x^8} d x= $

A

$\frac{\left(\tan ^{-1}\left(x^4\right)\right)^2}{8}+c$

B

$\frac{\left(\left(\tan ^{-1}\left(x^4\right)\right)^3\right.}{3}+c$

C

$\frac{\left(\tan ^{-1}\left(x^4\right)\right)^2}{4}+c$

D

$\frac{\left(\tan ^{-1}(x)^4\right)^2}{2}+c$

Question 71: Palette 71/138

AP-EAPCET 2024 MCQ

$ \int \frac{2}{1+x+x^2} d x= $

A

$\frac{4}{\sqrt{3}} \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)+c$

B

$\frac{4}{\sqrt{3}} \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)+c$

C

$\frac{2}{\sqrt{3}} \tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)+c$

D

$\frac{2}{\sqrt{3}} \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)+c$

Question 72: Palette 72/138

AP-EAPCET 2024 MCQ

$ \int \frac{1}{x^2\left(\sqrt{1+x^2}\right)} d x= $

A

$\frac{-\sqrt{x^2+1}}{x}+c$

B

$\frac{\sqrt{x^2+1}}{x}+c$

C

$\frac{-\sqrt{x^2-1}}{x}+c$

D

$\frac{\sqrt{x^2-1}}{x}+c$

Question 73: Palette 73/138

AP-EAPCET 2024 MCQ

$ \int \frac{\sin 7 x}{\sin 2 x \sin 5 x} d x= $

A

$\log (\sin 5 x \sin 2 x)+c$

B

$\log (\sin 5 x)+\log (\sin 2 x)+c$

C

$\frac{1}{5} \log (\sin 5 x)+\frac{1}{2} \log (\sin 2 x)+c$

D

$\frac{1}{5} \log (\sin x)+\frac{1}{2} \log (\sin x)+c$

Question 74: Palette 74/138

AP-EAPCET 2024 MCQ

If $\frac{x+2}{\left(x^2+3\right)\left(x^4+x^2\right)\left(x^2+2\right)}=\frac{A x+B}{x^2+3}+\frac{C x+D}{x^2+2}$ $+\frac{E x^3+F x^2+G x+H}{x^4+x^2}$, then $(E+F)(C+D)(A)=$

A

$-\frac{1}{4}$

B

$-\frac{3}{4}$

C

$\frac{3}{4}$

D

$\frac{1}{4}$

Question 75: Palette 75/138

AP-EAPCET 2024 MCQ

$\int \frac{\sin ^6 x}{\cos ^8 x} d x=$

A

$\tan 7 x+c$

B

$\frac{\tan ^7 x}{7}+c$

C

$\frac{\tan 7 x}{7}+c$

D

$\sec ^7 x$

Question 76: Palette 76/138

AP-EAPCET 2024 MCQ

$\int \frac{x^5}{x^2+1} d x=$

A

$\frac{x^4}{4}+\frac{x^3}{3}-\tan ^{-1} x+c$

B

$\frac{x^4}{4}-\frac{x^2}{2}+\frac{1}{2} \log \left(x^2+1\right)+c$

C

$\frac{x^4}{4}+\frac{x^3}{3}+\tan ^{-1} x+c$

D

$\frac{x^4}{4}+\frac{x^2}{2}-\frac{1}{2} \log \left(x^2+1\right)+c$

Question 77: Palette 77/138

AP-EAPCET 2024 MCQ

$\int {\left( {\sum\limits_{r = 0}^\infty {{{{x^r}{3^r}} \over {r!}}} } \right)dx = } $

A

$e^x+c$

B

$\frac{e^{3 x}}{3}+c$

C

$3 e^{3 x}+c$

D

$3 e^x+c$

Question 78: Palette 78/138

AP-EAPCET 2024 MCQ

$\int \frac{x^4+1}{x^6+1} d x=$

A

$\tan ^{-1} x-\tan ^{-1} x^3+c$

B

$\tan ^{-1} x-\frac{1}{3} \tan ^{-1} x^3+c$

C

$\tan ^{-1} x+\tan ^{-1} x^3+c$

D

$\tan ^{-1} x+\frac{1}{3} \tan ^{-1} x^3+c$

Question 79: Palette 79/138

AP-EAPCET 2024 MCQ

$\int e^x(x+1)^2 d x=$

A

$x e^x+c$

B

$e^x x^2+c$

C

$e^x\left(x^2+1\right)+c$

D

$e^x(x+1)+c$

Question 80: Palette 80/138

AP-EAPCET 2024 MCQ

If $\frac{1}{(3 x+1)(x-2)}=\frac{A}{3 x+1}+\frac{B}{x-2}$ and $\frac{x+1}{(3 x+1)(x-2)}=\frac{C}{3 x+1}+\frac{D}{x-2}$, then

A

$A+3 B=0, A: C=1: 3, B: D=2: 3$

B

$A+3 B=0, A: C=3: 1, B: D=3: 2$

C

$A-3 B=0, A: C=3: 2, B: D=1: 3$

D

$A+3 B=0, A: C=3: 2, B: D=1: 3$

Question 81: Palette 81/138

AP-EAPCET 2024 MCQ

If $x \in\left[2 n \pi-\frac{\pi}{4}, 2 n \pi+\frac{3 \pi}{4}\right]$ and $n \in Z$, then $\int \sqrt{1-\sin 2 x} d x=$

A

$-\cos x+\sin x+c$

B

$\cos x+\sin x+c$

C

$-\cos x-\sin x+c$

D

$\cos x-\sin x+c$

Question 82: Palette 82/138

AP-EAPCET 2024 MCQ

$\int e^x\left(\frac{x+2}{x+4}\right)^2 d x=$

A

$-\frac{x e^x}{(x+4)^2}+c$

B

$-\frac{x e^x}{(x+4)}+c$

C

$\frac{x e^x}{(x+4)}+c$

D

$\frac{2 x e^x}{(x+4)}+c$.

Question 83: Palette 83/138

AP-EAPCET 2024 MCQ

If $\int \frac{1}{1-\cos x} d x=\tan \left(\frac{x}{\alpha}+\beta\right)+c$, then one of the values of $\frac{\pi \alpha}{4}-\beta$ is

A

$-\frac{\pi}{2}$

B

$\pi$

C

0

D

$\frac{\pi}{4}$

Question 84: Palette 84/138

AP-EAPCET 2024 MCQ

If $n \geq 2$ is a natural number and $0<\theta<\frac{\pi}{2}$, then $\int \frac{\left(\cos ^n \theta-\cos \theta\right)^{1 / n}}{\cos ^{n+1} \theta} \sin \theta d \theta=$

A

$\frac{n}{n-1}\left(\cos ^{(1-n)} \theta-1\right)^2+c$

B

$\frac{n}{(n+1)(1-n)}\left(\cos ^{(1-n)} \theta-1\right)^{1+\frac{1}{n}}+c$

C

$\frac{1}{n-1}\left(\cos ^{(n-1)} \theta-1\right)^2+c$

D

$\frac{n}{1-n^2}\left(1-\cos ^{(1-n)} \theta\right)^{(n+1) / n}+c$

Question 85: Palette 85/138

AP-EAPCET 2024 MCQ

If $\frac{x^2+3}{x^4+2 x^2+9}=\frac{A x+B}{x^2+a x+b}+\frac{C x+D}{x^2+c x+b}$, then $a A+b B+c C+D=$

A

1

B

0

C

-1

D

2

Question 86: Palette 86/138

AP-EAPCET 2024 MCQ

$\int \frac{d x}{x\left(x^4+1\right)}=$

A

$\log \left(\frac{x}{x^4+1}\right)+c$

B

$\frac{3}{4} \log \left(x^4+1\right)+c$

C

$\frac{1}{3} \log \left(\frac{x^3}{x^4+1}\right)+c$

D

$\frac{1}{4} \log \left(\frac{x^4}{x^4+1}\right)+c$

Question 87: Palette 87/138

AP-EAPCET 2024 MCQ

$\int \frac{d x}{\sqrt{\sin ^3 x \cos (x-a)}}=$

A

$\frac{1}{\cos \alpha} \sqrt{\cot x+\tan \alpha}+c$

B

$\frac{1}{\sqrt{\cos \alpha}} \sqrt{\cot x-\tan \alpha}+c$

C

$\frac{-1}{\sqrt{\sin \alpha}} \sqrt{\cot x+\tan \alpha}+c$

D

$\frac{-2}{\sqrt{\cos \alpha}} \sqrt{\cot x+\tan \alpha}+c$

Question 88: Palette 88/138

AP-EAPCET 2024 MCQ

$\int \frac{e^{2 x}}{\sqrt[4]{e^x+1}} d x=$

A

$\frac{4}{7}\left(e^x+1\right)^{\frac{4}{3}}\left(3 e^x-1\right)+c$

B

$\frac{2}{21}\left(e^x+1\right)^{\frac{3}{4}}\left(3 e^x-7\right)+c$

C

$\frac{4}{21}\left(e^x+1\right)^{3 / 4}\left(3 e^x-4\right)+c$

D

$\frac{8}{21}\left(e^x+1\right)^{3 / 4}\left(3 e^x-1\right)+c$

Question 89: Palette 89/138

AP-EAPCET 2024 MCQ

$\int \frac{2-\sin x}{2 \cos x+3} d x=$

A

$\frac{2}{\sqrt{5}} \tan ^{-1}\left(\frac{1}{\sqrt{3}} \tan \frac{x}{2}\right)-\log \sqrt{2 \cos x+3}+c$

B

$\frac{4}{\sqrt{5}} \tan ^{-1}\left(\frac{1}{\sqrt{5}} \tan \frac{x}{2}\right)+\log \sqrt{2 \cos x+3}+c$

C

$\frac{3}{\sqrt{5}} \tan ^{-1}\left(\frac{1}{\sqrt{5}} \tan \frac{x}{2}\right)+\log \sqrt{2 \cos x-3}+c$

D

$\frac{1}{\sqrt{5}} \tan ^{-1}\left(\frac{1}{\sqrt{5}} \tan \frac{x}{3}\right)-\log \sqrt{2 \cos x-3}+c$

Question 90: Palette 90/138

AP-EAPCET 2024 MCQ

$\int \sin ^{-1} \sqrt{\frac{x}{a+x}} d x=$

A

$(a+x) \tan ^{-1} \sqrt{\frac{x}{a}}-\sqrt{a x}+c$

B

$\frac{1}{a+x} \tan ^{-1}\left(\frac{x}{a}\right)-\sqrt{a x}+c$

C

$(a+x) \tan ^{-1}\left(\frac{a}{x}\right)+\sqrt{a x}+c$

D

$\sqrt{a+x} \tan ^{-1}\left(\frac{x}{a}\right)+a x+c$

Question 91: Palette 91/138

AP-EAPCET 2024 MCQ

If $\frac{A}{x-a}+\frac{B x+C}{x^2+b^2}=\frac{1}{(x-a)\left(x^2+b^2\right)}$, then $\mathrm{C}=$

A

$\frac{-1}{a^2+b^2}$

B

$\frac{1}{a^2+b^2}$

C

$\frac{-a}{a^2+b^2}$

D

$\frac{a}{a^2+b^2}$

Question 92: Palette 92/138

AP-EAPCET 2024 MCQ

$\int \frac{2 x^2-3}{\left(x^2-4\right)\left(x^2+1\right)} d x=A \tan ^{-1} x+B \log (x-2)+C \log (x+2)$, then $6 A+7 B-5 C=$

A

9

B

10

C

6

D

8

Question 93: Palette 93/138

AP-EAPCET 2024 MCQ

$\int \frac{3 x^9+7 x^8}{\left(x^2+2 x+5 x^8\right)^2} d x=$

A

$\frac{x^7}{5 x^7+x+2}+c$

B

$\frac{x^7}{2\left(5 x^7+x+2\right)}+c$

C

$\frac{1}{2\left(5 x^7+x+2\right)}+c$

D

$\frac{-x^7}{2\left(5 x^7+x+2\right)}+c$

Question 94: Palette 94/138

AP-EAPCET 2024 MCQ

$\int \frac{\cos x+x \sin x}{x(x+\cos x)} d x=$

A

$\log \left|x^2+x \cos x\right|+c$

B

$\log \left|\frac{x}{x+\cos x}\right|+c$

C

$\log \left|\frac{\cos x}{x+\cos x}\right|+c$

D

$\log \left|\frac{1}{x+\cos x}\right|-\log x+c$

Question 95: Palette 95/138

AP-EAPCET 2024 MCQ

If $\int \sqrt{\frac{2}{1+\sin x}} d x=2 \log |A(x)-B(x)|+C$ and $0 \leq x \leq \frac{\pi}{2}$, then $B\left(\frac{\pi}{4}\right)=$

A

$\frac{1}{\sqrt{2+3 \sqrt{3}}}$

B

$\frac{1}{\sqrt{3+2 \sqrt{2}}}$

C

$\frac{-1}{\sqrt{3+2 \sqrt{2}}}$

D

$\frac{2}{\sqrt{2+\sqrt{2}}}$

Question 96: Palette 96/138

AP-EAPCET 2024 MCQ

$ \begin{aligned} &\text { If } \int \frac{3}{2 \cos ^3 x \sqrt{2 \sin 2 x}} d x=\frac{3}{2}(\tan x)^B+\frac{3}{10}(\tan x)^A+C \text {, than }\\&A= \end{aligned} $

A

$\frac{1}{2}$

B

1

C

5

D

$\frac{5}{2}$

Question 97: Palette 97/138

AP-EAPCET 2024 MCQ

If $\frac{1}{x^4+1}=\frac{A x+B}{x^2+\sqrt{2} x+1}+\frac{C x+D}{x^2-\sqrt{2} x+1}$, then $B D-A C=$

A

$\frac{3}{8}$

B

$\frac{1}{8}$

C

1

D

0

Question 98: Palette 98/138

AP-EAPCET 2024 MCQ

$ \int \frac{2 x^2 \cos x^2-\sin x^2}{x^2} d x= $

A

$\frac{\sin x^2}{x^2}+c$

B

$\frac{\cos x^2}{x^2}+c$

C

$\sin x^2+c$

D

$\frac{\sin x^2}{x}+c$

Question 99: Palette 99/138

AP-EAPCET 2024 MCQ

If $\int \frac{\log \left(1+x^4\right)}{x^3} d x=f(x) \log \left(\frac{1}{g(x)}\right)+\tan ^{-1}$ $(h(x))+c$, then $h(x)\left[f(x)+f\left(\frac{1}{x}\right)\right]=$

A

$h(x) g(-x)$

B

$\frac{g(x)}{2}$

C

$g(x)+g(-x)$

D

$g(x) h(x)$

Question 100: Palette 100/138

AP-EAPCET 2024 MCQ

Let $f(x)=\int \frac{x}{\left(x^2+1\right)\left(x^2+3\right)} d x$. If $f(3)=\frac{1}{4} \log \left(\frac{5}{6}\right)$, then $f(0)=$

A

$\frac{1}{4} \log \left(\frac{1}{3}\right)$

B

0

C

$\frac{1}{2} \log \left(\frac{1}{3}\right)$

D

$\log \left(\frac{1}{3}\right)$

Question 101: Palette 101/138

AP-EAPCET 2024 MCQ

$ \int \frac{2 \cos 2 x}{(1+\sin 2 x)(1+\cos 2 x)} d x= $

A

$2 \tan x+\log (1+\tan x)+c$

B

$\tan x-2 \log (1+\tan x)+c$

C

$2 \log (1+\tan x)+\tan x+c$

D

$2 \log (1+\tan x)-\tan x+c$

Question 102: Palette 102/138

AP-EAPCET 2024 MCQ

$ \int\left(\frac{x}{x \cos x-\sin x}\right)^2 d x= $

A

$\frac{x \operatorname{cosec} x}{x \cos x-\sin x}+\cot x+c$

B

$\frac{x \operatorname{cosec} x}{x \cos x-\sin x}-\cot x+c$

C

$\frac{x \operatorname{cosec} x}{x \cos +\sin x}+\cot x+c$

D

$\frac{x}{x \cos x-\sin x}-\cot x+c$

Question 103: Palette 103/138

AP-EAPCET 2024 MCQ

$\int \frac{1}{x^5 \sqrt[3]{x^3+1}} d x=$

A

$\frac{4}{\sqrt{x^5+1}}+c$

B

$4 x^4\left(x^5+1\right)^{4 / 3}+0$

C

$=\frac{\left(x^3+1\right)^{4 / 3}}{4 x^4}+c$

D

$-\frac{\left(x^5+1\right)^{45}}{4 x^5}+c$

Question 104: Palette 104/138

AP-EAPCET 2024 MCQ

$\int \frac{x+1}{\sqrt{x^2+x+1}} d x=$

A

$\frac{1}{2} \sqrt{x^2+x+1}+\frac{1}{2} \cosh ^{-1}\left(\frac{x+2}{\sqrt{3}}\right)+c$

B

$\frac{1}{2} \sqrt{x^2+x+1}+\frac{2}{\sqrt{3}} \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)+c$

C

$\sqrt{x^2+x+1}+\frac{2}{\sqrt{3}} \log \left|x^2+x+1\right|+c$

D

$\sqrt{x^2+x+1}+\frac{1}{2} \sinh ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)+c$

Question 105: Palette 105/138

AP-EAPCET 2024 MCQ

$\int\left(\tan ^9 x+\tan x\right) d x=0$

A

$\frac{\tan ^2 x}{12}\left(2 \tan ^3 x-3 \tan ^2 x+6\right)+c$

B

$\frac{\tan ^2 x}{6}-\frac{\tan ^5 x}{4}+\frac{\tan ^2 x}{2}+c$

C

$\left.\frac{\tan ^2 x^2}{6} \tan ^4 x+3 \tan ^2 x+4\right)+c$

D

$\frac{\tan x}{12} \tan ^4 x-3 \tan ^2 x+6+c$

Question 106: Palette 106/138

AP-EAPCET 2024 MCQ

$\int \frac{\operatorname{cosec} x}{3 \cos x+4 \sin x} d x=$

A

$\frac{1}{2} \log \left|\frac{\cos x}{3 \sin x+4 \cos x}\right|+c$

B

$\frac{1}{3} \log \left|\frac{\sin x}{3 \cos x+4 \sin x}\right|+c$

C

$\frac{1}{3} \log \left|\frac{3 \cos x+\sin x}{3 \cos x+4 \sin x}\right|+c$

D

$\frac{1}{2} \log \left|\frac{\cos x+4 \sin x}{3 \cos x+4 \sin x}\right|+c$

Question 107: Palette 107/138

AP-EAPCET 2024 MCQ

$\int e^{2 x+3} \sin 6 x d x=$

A

$\frac{e^{2 x+3}}{40}(2 \sin (x x+6 \cos 6 x)+c$

B

$\frac{e^{2 v+3}}{40}(2 \cos 6 x+6 \sin 6 x)+c$

C

$\frac{e^{2 n+3}}{20}(\sin 6 x-3 \cos 6 x)+c$

D

$\frac{e^{2 n 3}}{20}(\cos 8 x-3 \sin 6 x)+c$

Question 108: Palette 108/138

AP-EAPCET 2022 MCQ

$\frac{2 x^2+1}{x^3-1}=\frac{A}{x-1}+\frac{B x+C}{x^2+x+1} \Rightarrow 7 A+2 B+C=$

A

8

B

9

C

10

D

11

Question 109: Palette 109/138

AP-EAPCET 2022 MCQ

$\int \frac{3 x+4}{x^3-2 x+4} d x=\log f(x)+C \Rightarrow f(3)=$

A

$\frac{1}{\sqrt{17}}$

B

$\frac{1}{17}$

C

$\frac{2}{15}$

D

$\frac{2}{17}$

Question 110: Palette 110/138

AP-EAPCET 2022 MCQ

$\int \frac{e^{\tan ^{-1} x}}{1+x^2}\left[\left(\sec ^{-1} \sqrt{1+x^2}\right)^2+\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right] d x=$

A

$ e^{\tan ^{-1} x}\left(\tan ^{-1} x\right)^2+C $

B

$ e^{\tan ^{-1} x}\left(\sec ^{-1} x\right)^2+C $

C

$ e^{\tan ^{-1} x}\left(\sec ^{-1}\left(\sqrt{1+x^2}\right)\right)+C $

D

$ e^{\tan ^{-1}} \times\left(\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right)+C $

Question 111: Palette 111/138

AP-EAPCET 2022 MCQ

$\int \frac{d x}{(x-3)^{\frac{4}{5}}(x+1)^{\frac{6}{5}}}=$

A

$\frac{5}{4} \sqrt[5]{\frac{x-3}{x+1}}+C$

B

$\frac{5}{4}\left(\frac{x+1}{x-3}\right)^{\frac{1}{5}}+C$

C

$\frac{1}{5}\left(\frac{x-3}{x+1}\right)^{\frac{1}{5}}+c$

D

$\frac{5}{4}\left(\frac{x-3}{x+4}\right)^{\frac{4}{5}}+C$

Question 112: Palette 112/138

AP-EAPCET 2022 MCQ

If $I_n=\int\left(\cos ^n x+\sin ^n x\right) d x$ and $I_n-\frac{n-1}{n} I_{n-2} =\frac{\sin x \cos x}{n} f(x)$, then $f(x)=$

A

$\cos ^{n-2} x+\sin ^{n-2} x$

B

$\cos ^{n-2} x-\sin ^{n-2} x$

C

$\frac{\cos ^{n-2} x-\sin ^{n-2} x}{n}$

D

$\frac{\cos ^{n-2} x+\sin ^{n-2} x}{n}$

Question 113: Palette 113/138

AP-EAPCET 2022 MCQ

If $f(x)=\int x^2 \cos ^2 x\left(2 x \tan ^2 x-2 x-6 \tan x\right) d x$ and $f(0)=\pi$, then $f(x)=$

A

$x^2 \sin x+\pi$

B

$\cos x+\pi-1$

C

$-x^3 \sin 2 x+\pi$

D

$x^3 \cos 2 x+\pi \cos x$

Question 114: Palette 114/138

AP-EAPCET 2022 MCQ

If $\int \frac{e^{\sqrt{x}}}{\sqrt{x}}(x+\sqrt{x}) d x=e^{\sqrt{x}}[A x+B \sqrt{x}+C]+K$ then $A+B+C=$

A

$-$2

B

2

C

4

D

$-$4

Question 115: Palette 115/138

AP-EAPCET 2022 MCQ

If $\int \frac{1+\sqrt{\tan x}}{\sin 2 x} d x=A \log \tan x+B \tan x+C$, then $4 A-2 B=$

A

$-$1

B

2

C

1

D

$-$2

Question 116: Palette 116/138

AP-EAPCET 2022 MCQ

$\int \frac{1+\tan x \tan (x+a)}{\tan x \tan (x+a)} d x=$

A

$\tan a(\log (\sec (x+a))+\log \sec x+C$

B

$\cot a(\log |\sin x|-\log |\sin (x+a)|)+C$

C

$\tan a\left(\log \left(\frac{\cos x}{\sin (x+a)}\right)\right)+C$

D

$\cot a\left(\log \frac{\sin (x+a)}{\cos (x+a)}\right)+C$

Question 117: Palette 117/138

AP-EAPCET 2022 MCQ

Assertion (A) If $I_n=\int \cot ^n x d x$, then $I_6+I_4=\frac{-\cot ^5 x}{5}$

Reason (R) $\int \cot ^n x d x=\frac{-\cot ^{n-1} x}{n} -\int \cot ^{n-2} x d x$

A

A is false, R is false

B

A is true, R is true

C

A is true, R is false

D

A is false, R is true

Question 118: Palette 118/138

AP-EAPCET 2022 MCQ

If $I_n=\int \tan ^n x d x$, and $I_0+I_1+2 I_2+2 I_3+2 I_4 +I_5+I_6=\sum_\limits{k=1}^n \frac{\tan ^k x}{k}$, then $n=$

A

6

B

5

C

4

D

3

Question 119: Palette 119/138

AP-EAPCET 2022 MCQ

$\int \frac{e^{\cot x}}{\sin ^2 x}(2 \log \operatorname{cosec} x+\sin 2 x) d x=$

A

$-2 e^{\cot x} \log \left(\operatorname{cosec}^2 x\right)+C$

B

$-2 e^{\cot x} \log (\operatorname{cosec} x)+C$

C

$-2 e^{\cot x} \log (\operatorname{cosec} x+\sin x)+C$

D

$-2 e^{\cot x} \log (\operatorname{cosec} x-\cot x)+C$

Question 120: Palette 120/138

AP-EAPCET 2022 MCQ

The parametric form of a curve is $x=\frac{t^3}{t^2-1} y=\frac{t}{t^2-1}$, then $\int \frac{d x}{x-3 y}=$

A

$\frac{1}{2} \log \left(t^2-1\right)+C$

B

$2 \log \left(t\left(t^2-1\right)\right)+C$

C

$\frac{1}{4} \log \left(\frac{t}{t^2-3}\right)+C$

D

$\frac{5}{2} \log \left(t+\frac{1}{t^2}\right)+C$

Question 121: Palette 121/138

AP-EAPCET 2021 MCQ

If

$\begin{aligned} \frac{2 x^4-x^3+3 x^2-x+4}{x^2-3 x+2} =f(x)+\frac{A}{x-1}+\frac{B}{x-2}\end{aligned}$, then

A

$f(x)=2 x^2+5 x+14, A+B=39$

B

$f(x)=2 x^2-5 x+14, A+B=31$

C

$f(x)=2 x^2+5 x+14, A+B=31$

D

$f(x)=2 x^2+5 x+14, A=4, B=35$

Question 122: Palette 122/138

AP-EAPCET 2021 MCQ

If $f^{\prime}(x)=x+\frac{1}{x}$, then $f(x)$ is equal to

A

$x^2+\log (x)+c$

B

$\frac{x^2}{2}+\log (x)+c$

C

$x+\log (x)+c$

D

$\frac{x}{2}+\log (x)+c$

Question 123: Palette 123/138

AP-EAPCET 2021 MCQ

If $f(x)=\frac{1}{\left(\cos ^2 x\right) \sqrt{1+\tan x}}$, then its antiderivative $F(x)=$ ........, given, $F(0)=4$

A

$\sqrt{1+\tan x}+4$

B

$\frac{2}{3}(1+\tan x)^{\frac{3}{2}}$

C

$2(\sqrt{1+\tan x}+1)$

D

$\sqrt{1+\tan x}+2$

Question 124: Palette 124/138

AP-EAPCET 2021 MCQ

If the primitive of $\cos (\log x)$ is $f(x)\{\cos (g(x))+\sin (h(x))\}$, then which among the following is true?

A

$h^{\prime}(x)=\frac{-1}{x}$

B

$f^{\prime}(x)=\frac{1}{2}$

C

$g^{\prime}(x)=\log (x)$

D

$h(x)=\frac{x}{2}$

Question 125: Palette 125/138

AP-EAPCET 2021 MCQ

$\int \frac{\sec x}{\sqrt{\sin (2 x+\theta)+\sin \theta}} d x$ is equal to

A

$\sqrt{(\tan x+\tan \theta) \sec \theta}+c$

B

$\sqrt{2(\tan x+\tan \theta) \sec \theta}+c$

C

$\sqrt{2(\sin x+\tan \theta) \sec \theta}+c$

D

$\sqrt{2(\cos x+\tan \theta) \sec \theta}+c$

Question 126: Palette 126/138

AP-EAPCET 2021 MCQ

Given, $\frac{3 x-2}{(x+1)^2(x+3)}=\frac{A}{x+1} +\frac{B}{(x+1)^2}+\frac{C}{x+3}$, then $4 A+2 B+4 C$ is equal to

A

5

B

$-$5

C

$-$3

D

3

Question 127: Palette 127/138

AP-EAPCET 2021 MCQ

$\int \frac{\sin \alpha}{\sqrt{1+\cos \alpha}} d \alpha$ is equal to

A

$-2 \sqrt{2} \cos \left(\frac{\alpha}{2}\right)+C$

B

$2 \sqrt{2} \cos \left(\frac{\alpha}{2}\right)+C$

C

$\sqrt{2} \cos \left(\frac{\alpha}{2}\right)+C$

D

$-\sqrt{2} \cos \left(\frac{\alpha}{2}\right)+C$

Question 128: Palette 128/138

AP-EAPCET 2021 MCQ

If $\int \frac{\cos 4 x+1}{\cot x-\tan x}=k \cos 4 x+C$, then $k$ is equal to

A

$\frac{-1}{2}$

B

$\frac{-1}{8}$

C

$\frac{-1}{3}$

D

$\frac{-1}{5}$

Question 129: Palette 129/138

AP-EAPCET 2021 MCQ

If $\int\left[\cos (x) \cdot \frac{d}{d x}(\operatorname{cosec}(x)] d x=f(x)+g(x)+c\right.$ then $f(x) \cdot g(x)$ is equal to

A

$x \cot (x)$

B

$x \tan (x)$

C

$x \cos (x)$

D

1

Question 130: Palette 130/138

AP-EAPCET 2021 MCQ

If $\int \frac{(2 x+1)^6}{(3 x+2)^8} d x=P\left(\frac{2 x+1}{3 x+2}\right)^Q+R$, then $\frac{P}{Q}$ is equal to

A

$\frac{1}{7^2}$

B

$\frac{1}{7}$

C

$7^2$

D

7

Question 131: Palette 131/138

AP-EAPCET 2021 MCQ

Which of the following is partial fraction of $\frac{-x^2+6 x+13}{(3 x+5)\left(x^2+4 x+4\right)}$ is equal to

A

$\frac{3}{3 x+5}+\frac{-1}{x+2}+\frac{2}{(x+2)^2}$

B

$\frac{2}{3 x+5}+\frac{-1}{x+2}+\frac{3}{(x+2)^2}$

C

$\frac{-1}{3 x+5}+\frac{2}{x+2}+\frac{3}{(x+2)^2}$

D

$\frac{3}{3 x+5}+\frac{2}{x+2}+\frac{-1}{(x+2)^2}$

Question 132: Palette 132/138

AP-EAPCET 2021 MCQ

$\int \frac{1+x+\sqrt{x+x^2}}{\sqrt{x}+\sqrt{1+x}} d x$ is equal to

A

$\frac{1}{2}(\sqrt{1+x})+c$

B

$\sqrt{1+x}+c$

C

$2(1+x)^{3 / 2}+c$

D

$\frac{2}{3}(1+x)^{3 / 2}+c$

Question 133: Palette 133/138

AP-EAPCET 2021 MCQ

$\int(\cos x) \log \cot \left(\frac{x}{2}\right) d x$ is equal to

A

$(\sin x) \log \cot \left(\frac{x}{2}\right)+c$

B

$(\cos x) \log \cot \left(\frac{x}{2}\right)+c$

C

$(\sin x) \log \cot \left(\frac{x}{2}\right)+x+c$

D

$(\sin x) \log \cot \left(\frac{x}{2}\right)-x+c$

Question 134: Palette 134/138

AP-EAPCET 2021 MCQ

$\int \sqrt{e^{4 x}+e^{2 x}} d x$ is equal to

A

$\frac{1}{2} e^x\left(\sqrt{e^{2 x}+1}\right)+\frac{1}{2} \sinh ^{-1}\left(e^x\right)+c$

B

$\frac{1}{2} e^x\left(\sqrt{e^{2 x}+1}\right)+\sinh ^{-1}\left(e^x\right)+c$

C

$\frac{1}{2}\left(\sqrt{e^{2 x}+1}\right)+\frac{1}{2} \sinh ^{-1}\left(e^x\right)+c$

D

$\sqrt{e^{4 x}+e^{2 x}}+\sqrt{e^{2 x}+1}+c$

Question 135: Palette 135/138

AP-EAPCET 2021 MCQ

If $\int \frac{1}{1+\sin x} d x=\tan (f(x))+c$, then $f^{\prime}(0)$ is equal to

A

0

B

1

C

$\frac{1}{2}$

D

$\frac{-1}{2}$

Question 136: Palette 136/138

AP-EAPCET 2021 MCQ

$\int \frac{e^x(x+3)}{(x+5)^3} d x$ is equal to

A

$\frac{e^x}{(x+5)^2}+C$

B

$e^x(x+5)^2+C$

C

$e^x(x+3)^2+C$

D

$\frac{e^x}{(x+3)^2}+C$

Question 137: Palette 137/138

AP-EAPCET 2021 MCQ

If $\int \frac{(x-1)^2}{\left(x^2+1\right)^2} d x=\tan ^{-1}(x)+g(x)+k$, then $g(x)$ is equal to

A

$\tan ^{-1}\left(\frac{x}{2}\right)$

B

$\frac{1}{x^2+1}$

C

$\frac{1}{2\left(x^2+1\right)}$

D

$\frac{2}{x^2+1}$

Question 138: Palette 138/138

AP-EAPCET 2021 MCQ

If $\int \frac{1-(\cot x)^{2021}}{\tan x+(\cot x)^{2022}} d x=\frac{1}{A} \log\left|(\sin x)^{2023}+(\cos x)^{2023}\right|+c$, then $A$ is equal to

A

2020

B

2021

C

2022

D

2023