Question 1: Palette 1/82

AP-EAPCET 2025 MCQ

$ \int_0^1 x \sin ^{-1} x d x= $

A

$\frac{\pi}{8}$

B

$\frac{\pi}{4}$

C

$\frac{\pi}{12}$

D

$\frac{\pi}{3}$

Question 2: Palette 2/82

AP-EAPCET 2025 MCQ

$ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin (x-[x]) d x= $

Here $[x]$ is the greatest integer function

A

0

B

$3(1-\cos 1)+\sin 2-\sin 1$

C

$3(1-\cos 1)+\cos 2-\sin 1$

D

$\cos 2-\sin 2$

Question 3: Palette 3/82

AP-EAPCET 2025 MCQ

$ \int_0^2 x^2(2-x)^5 d x= $

A

$\frac{128}{21}$

B

$\frac{64}{7}$

C

$\frac{32}{21}$

D

$\frac{16}{7}$

Question 4: Palette 4/82

AP-EAPCET 2025 MCQ

If $f(x)=\max \left\{x^3-4, x^4-4\right\}$ and $g(x)=\min \left\{x^2, x^3\right\}$, then $\int_{-1}^1(f(x)-g(x)) d x=$

A

$-\frac{151}{20}$

B

$\frac{9}{20}$

C

$\frac{131}{22}$

D

$-\frac{67}{9}$

Question 5: Palette 5/82

AP-EAPCET 2025 MCQ

$ \int_0^1 \frac{2 x+5}{x^2+3 x+2} d x= $

A

$\log \left(\frac{16}{3}\right)$

B

0

C

$\log \left(\frac{3}{16}\right)$

D

$4 \log 2-2 \log 3$

Question 6: Palette 6/82

AP-EAPCET 2025 MCQ

$ \int_0^1 x^{\frac{5}{2}}(1-x)^{\frac{3}{2}} d x= $

A

$\frac{5 \pi}{256}$

B

$\frac{3 \pi}{256}$

C

$\frac{3 \pi}{128}$

D

$\frac{5 \pi}{128}$

Question 7: Palette 7/82

AP-EAPCET 2025 MCQ

$ \lim _{n \rightarrow \infty}\left[\begin{array}{c} \frac{1}{n^2} \sec ^2 \frac{1}{n^2}+\frac{2}{n^2} \sec ^2 \frac{4}{n^2}+\frac{3}{n^2} \sec ^2 \\ \frac{9}{n^2}+\ldots+\frac{1}{n^2} \sec ^2 1 \end{array}\right]= $

A

$\tan ^{-1} 1$

B

$\frac{1}{2} \tan ^{-1} 1$

C

$\frac{1}{2} \tan 1$

D

$\frac{1}{2} \sec 1$

Question 8: Palette 8/82

AP-EAPCET 2025 MCQ

$ \int_0^\pi\left(\sin ^5 x \cos ^3 x+\sin ^4 x \cos ^4 x+\sin ^3 x \cos ^4 x\right) d x= $

A

$\frac{873}{2240}$

B

$\frac{3 \pi}{128}+\frac{12}{35}$

C

$\frac{1641}{4480}$

D

$\frac{3 \pi}{128}+\frac{4}{35}$

Question 9: Palette 9/82

AP-EAPCET 2025 MCQ

$ \int_0^1 \frac{x^4+1}{x^6+1} d x= $

A

$\frac{\pi}{3}$

B

$\frac{\pi}{4}$

C

$\frac{\pi}{6}$

D

$\frac{\pi}{2}$

Question 10: Palette 10/82

AP-EAPCET 2025 MCQ

$ \int_{-2 \pi}^{2 \pi} \sin ^4(2 x) \cos ^6(2 x) d x= $

A

$\frac{3 \pi}{64}$

B

$\frac{9 \pi}{64}$

C

$\frac{9 \pi}{35}$

D

$\frac{9 \pi}{280}$

Question 11: Palette 11/82

AP-EAPCET 2025 MCQ

If $f(t)=\int_0^t \tan ^{(2 n-1)} x d x, n \in N$, then $f(t+\pi)=$

A

$f(t) f(\pi)$

B

$f(t)-f(\pi)$

C

$f(t)+f(\pi)$

D

$\frac{f(t)}{f(\pi)}$

Question 12: Palette 12/82

AP-EAPCET 2025 MCQ

$ \int_0^2 x^8\left(\frac{4}{x^2}-1\right)^{\frac{5}{2}} d x= $

A

$\frac{2^{15}}{63}$

B

$\frac{2^{16}}{315}$

C

$\frac{2^{16}}{189}$

D

$\frac{2^{10}}{63}$

Question 13: Palette 13/82

AP-EAPCET 2025 MCQ

$ \int_{-\pi / 2}^{\pi / 2} \sin ^2 x \cos ^2 x(\sin x+\cos x) d x= $

A

0

B

$\frac{2}{15}$

C

$\frac{4}{15}$

D

$\frac{2}{5}$

Question 14: Palette 14/82

AP-EAPCET 2025 MCQ

$ \int_{1 / 5}^{1 / 2} \frac{\sqrt{x-x^2}}{x^3} d x= $

A

$\frac{21}{2}$

B

$\frac{14}{3}$

C

$\frac{7}{3}$

D

$\frac{7}{2}$

Question 15: Palette 15/82

AP-EAPCET 2025 MCQ

$ \int_0^{400 \pi} \sqrt{1-\cos 2 x} d x= $

A

$100 \sqrt{2}$

B

$200 \sqrt{2}$

C

$400 \sqrt{2}$

D

$800 \sqrt{2}$

Question 16: Palette 16/82

AP-EAPCET 2025 MCQ

$ \int_0^x \frac{t^2}{\sqrt{a^2+t^2}} d t= $

A

$\frac{x}{2} \sqrt{a^2+x^2}+\log \left|x+\sqrt{a^2+x^2}\right|$

B

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

C

$\frac{x}{2} \sqrt{a^2+x^2}+\frac{a^2}{4} \log \left|x+\sqrt{a^2+x^2}\right|$

D

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

Question 17: Palette 17/82

AP-EAPCET 2025 MCQ

$ \int_{\frac{5}{6}}^\pi \cos ^{-4} x d x= $

A

$\frac{64}{9 \sqrt{3}}$

B

$\frac{52 \sqrt{3}}{9}$

C

$\frac{62 \sqrt{3}}{9}$

D

$\frac{44}{9 \sqrt{3}}$

Question 18: Palette 18/82

AP-EAPCET 2025 MCQ

$ \int\limits_0^{\frac{3 \pi}{2}} \frac{\cos ^3 x}{\cos ^3 x+\sin ^3 x} d x= $

A

0

B

1

C

$\frac{\pi}{4}$

D

$\frac{3 \pi}{4}$

Question 19: Palette 19/82

AP-EAPCET 2025 MCQ

If $k \in N$, then $\lim\limits_{n \rightarrow \infty}\left[\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+\ldots .+\frac{1}{k n}\right]=$

A

$\log (k+1)$

B

$\log k$

C

$\log (k+5)$

D

$\log (k+1)-\log 6$

Question 20: Palette 20/82

AP-EAPCET 2025 MCQ

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

A

0

B

$\frac{\pi}{2}$

C

$\frac{3 \pi}{2}$

D

$\frac{5 \pi}{2}$

Question 21: Palette 21/82

AP-EAPCET 2025 MCQ

$ \int_0^{\pi / 4} \frac{\cos ^2 x}{\cos ^2 x+4 \sin ^2 x} d x= $

A

$\frac{\pi}{2}-\frac{1}{3} \tan ^{-1} 2$

B

$-\frac{\pi}{4}-\frac{4}{3} \tan ^{-1} 2$

C

$\frac{\pi}{6}+\frac{2}{3} \tan ^{-1} 2$

D

$-\frac{\pi}{12}+\frac{2}{3} \tan ^{-1} 2$

Question 22: Palette 22/82

AP-EAPCET 2025 MCQ

$ \int_{5 \pi}^{25 \pi}|\sin 2 x+\cos 2 x| d x= $

A

$20 \sqrt{2}$

B

$10 \sqrt{2}$

C

$40 \sqrt{2}$

D

$80 \sqrt{2}$

Question 23: Palette 23/82

AP-EAPCET 2025 MCQ

$\int_{\frac{-\pi}{4}}^{\frac{\pi}{3}}\left|\tan \left(x-\frac{\pi}{6}\right)\right| d x=$

A

$\log \frac{\sqrt{3}-1}{\sqrt{6}}$

B

$\log (2 \sqrt{2}(\sqrt{3}+1))$

C

$\log \frac{\sqrt{3}+1}{\sqrt{6}}$

D

$\log (2 \sqrt{2}(\sqrt{3}-1))$

Question 24: Palette 24/82

AP-EAPCET 2025 MCQ

$ \int_0^\pi \frac{x \sin x}{\sin ^2 x+2 \cos ^2 x} d x= $

A

$\frac{\pi}{2}$

B

$\frac{\pi^2}{2}$

C

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

D

$\frac{\pi}{4}$

Question 25: Palette 25/82

AP-EAPCET 2025 MCQ

$ \mathop {\lim }\limits_{n \to \infty }\left(\frac{1}{1^2+n^2}+\frac{2}{2^2+n^2}+\frac{3}{3^2+n^2}+\ldots+\frac{n}{n^2+n^2}\right)= $

A

1

B

$\frac{1}{2} \log 2$

C

$2 \log 2$

D

0

Question 26: Palette 26/82

AP-EAPCET 2025 MCQ

$ \int_0^{\frac{\pi}{2}} \log |\tan x+\cot x| d x= $

A

$\pi \log 2$

B

$-\pi \log 2$

C

$\frac{\pi}{2} \log 2$

D

$2 \pi \log 2$

Question 27: Palette 27/82

AP-EAPCET 2025 MCQ

$ \int_0^\pi x \cdot \sin ^5 x \cdot \cos ^6 x d x= $

A

$\frac{16 \pi}{693}$

B

$\frac{8 \pi}{693}$

C

$\frac{4 \pi}{693}$

D

$\frac{2 \pi}{693}$

Question 28: Palette 28/82

AP-EAPCET 2025 MCQ

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

A

$\log (\sqrt{3}+1)$

B

$\log (\sqrt{3}-1)$

C

$\log (3+\sqrt{3})$

D

$\log (3-\sqrt{3})$

Question 29: Palette 29/82

AP-EAPCET 2025 MCQ

Let $H(x)=3 x^4+6 x^3-2 x^2+1$ and $g(x)$ be a linear polynomial. If $\frac{H(x)}{(x-1)(x+1)(x-2)}=f(x) +\frac{g(x)}{(x-1)(x+1)(x-2)}$, then

$H(-1)+2 H(2)-3 H(1)=$

A

$f(-1)+2 f(2)-3 f(1)$

B

$H(-1)+f(2)+g(3)$

C

$g(-1)+2 g(2)-3 g(1)$

D

$H(1)+2 f(2)-g(1)$

Question 30: Palette 30/82

AP-EAPCET 2025 MCQ

$ \int_{\pi / 4}^{\pi / 3} \frac{\cos x-\sin x}{\sin 2 x} d x= $

A

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

B

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

C

$\log \left[\frac{(3-2 \sqrt{2})(2-\sqrt{3})}{\sqrt{3}}\right]$

D

$\log \left[\frac{(3+2 \sqrt{2})(2-\sqrt{3})}{\sqrt{3}}\right]$

Question 31: Palette 31/82

AP-EAPCET 2025 MCQ

$ \int_0^{\pi / 2} \frac{\sin x}{1+\cos x+\sin x} d x= $

A

$\frac{\pi}{2}+\frac{1}{2} \log 2$

B

$\frac{\pi}{4}-\frac{1}{2} \log 2$

C

$\frac{\pi}{4}$

D

$\frac{3 \pi}{4}+\log 2$

Question 32: Palette 32/82

AP-EAPCET 2025 MCQ

$ \int_0^\pi \frac{x \sin x}{1+\cos ^2 x} d x= $

A

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

B

$\frac{\pi}{2}$

C

$\frac{\pi^2}{2}$

D

$\frac{\pi}{4}$

Question 33: Palette 33/82

AP-EAPCET 2024 MCQ

If $\int_0^{2 \pi}\left(\sin ^4 x+\cos ^4 x\right) d x=K \int_0^\pi \sin ^2 x d x+L \int_0^{\frac{\pi}{2}} \cos ^2 x d x$ and $K, L \in N$, then the number of possible ordered pairs ( $K, L$ ) is

A

1

B

2

C

3

D

4

Question 34: Palette 34/82

AP-EAPCET 2024 MCQ

$\int_0^\pi \frac{x \sin x}{4 \cos ^2 x+3 \sin ^2 x} d x$ is equal to

A

$\frac{\pi^2}{6 \sqrt{3}}$

B

$\frac{\pi}{3 \sqrt{3}}$

C

$\frac{\pi^2}{3 \sqrt{3}}$

D

$\sqrt{3} \pi^2$

Question 35: Palette 35/82

AP-EAPCET 2024 MCQ

If $A=\int_0^{\infty} \frac{1+x^2}{1+x^4} d x, B=\int_0^1 \frac{1+x^2}{1+x^4} d x$, then

A

$2 A=B$

B

$A=B$

C

$2 B=A$

D

$2 B+A=0$

Question 36: Palette 36/82

AP-EAPCET 2024 MCQ

$\int_0^1 \sqrt{\frac{2+x}{2-x}} d x$ is equal to

A

$\pi+2$

B

$\frac{1}{2}(\pi+2)$

C

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

D

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

Question 37: Palette 37/82

AP-EAPCET 2024 MCQ

If $M=\int\limits_0^{\infty} \frac{\log t}{1+t^3} d t$ and $N=\int\limits_{-\infty}^{\infty} \frac{t e^{2 t}}{1+e^{3 t}} d t$, then

A

$N=2 M$

B

$N=M$

C

$N=3 M$

D

$N=-M$

Question 38: Palette 38/82

AP-EAPCET 2024 MCQ

$\int\limits_{-2}^2\left(4-x^2\right)^{\frac{5}{2}} d x$ is equal to

A

$40 \pi$

B

$20 \pi$

C

$10 \pi$

D

$\frac{5 \pi}{32}$

Question 39: Palette 39/82

AP-EAPCET 2024 MCQ

$ \mathop {\lim }\limits_{x \to \infty }\left[\left(1+\frac{1}{n^3}\right)^{\frac{1}{n^3}}\left(1+\frac{8}{n^3}\right)^{\frac{4}{n^3}}\left(1+\frac{27}{n^3}\right)^{\frac{9}{n^3}} \ldots . .(2)^{\frac{1}{n}}\right] \text { is equaln } $

A

$\log 2-\frac{1}{2}$

B

$e^{\left(\log 2-\frac{1}{2}\right)}$

C

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

D

$\frac{1}{3}(2 \log 2-1)$

Question 40: Palette 40/82

AP-EAPCET 2024 MCQ

$\int\limits_{-5 \pi}^{5 \pi}(1-\cos 2 x)^{\frac{5}{2}} d x$ is equal to

A

$\frac{64 \sqrt{2}}{5}$

B

$\frac{128 \sqrt{2}}{5}$

C

$\frac{256 \sqrt{2}}{3}$

D

$\frac{128 \sqrt{2}}{3}$

Question 41: Palette 41/82

AP-EAPCET 2024 MCQ

$ \int_0^{\pi / 4} \log (1+\tan x) d x= $

A

$\pi \log 2+1$

B

$\frac{\pi}{2} \log 2+1$

C

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

D

$\frac{\pi}{8} \log 2$

Question 42: Palette 42/82

AP-EAPCET 2024 MCQ

$\int\limits_\pi ^\pi {}\frac{x \sin x}{1+\cos ^2 x} d x= $

A

$\frac{3 \pi^2}{4}$

B

$\frac{\pi}{2}+1$

C

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

D

$\frac{\pi^2}{2}$

Question 43: Palette 43/82

AP-EAPCET 2024 MCQ

$\int\limits_0^{\pi /4} {{{{x^2}} \over {{{(x\,\sin \,x + \cos \,x)}^2}}}dx = } $

A

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

B

$\frac{4-\pi}{4+\pi}$

C

$\frac{6-\pi}{6+\pi}$

D

$\frac{8-\pi}{8+\pi}$

Question 44: Palette 44/82

AP-EAPCET 2024 MCQ

$\int_0^1 \frac{x}{(1-x)^{\frac{3}{4}}} d x=$

A

$\frac{4}{5}$

B

$\frac{8}{15}$

C

$\frac{14}{5}$

D

$\frac{16}{5}$

Question 45: Palette 45/82

AP-EAPCET 2024 MCQ

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

A

2

B

4

C

0

D

8

Question 46: Palette 46/82

AP-EAPCET 2024 MCQ

$\int_1^5(|x-3|+|1-x|) d x=$

A

4

B

8

C

12

D

24

Question 47: Palette 47/82

AP-EAPCET 2024 MCQ

If $729 \int_1^3 \frac{1}{x^3\left(x^2+9\right)^2} d x=a+\log b$, then $(a-b)=$

A

4

B

$-\frac{4}{5}$

C

$\frac{4}{5}$

D

-4

Question 48: Palette 48/82

AP-EAPCET 2024 MCQ

$\lim \limits_{n \rightarrow \infty} \frac{1^{17}+2^{77}+\ldots+n^{77}}{n^{78}}=$

A

$\frac{1}{77}$

B

1

C

76

D

$\frac{1}{78}$

Question 49: Palette 49/82

AP-EAPCET 2024 MCQ

$ \text { If } f(x)=\left\{\begin{array}{cc} \frac{6 x^2+1}{4 x^3+2 x+3} & , 0 < x < 1 \\ x^2+1 & , 1 \leq x < 2 \end{array} \text {, then } \int_0^2 f(x) d x=\right. $

A

$\frac{1}{2} \log 3+\frac{10}{3}$

B

$\frac{1}{2} \log 3-\frac{10}{3}$

C

$\frac{1}{2} \log 3+\frac{13}{3}$

D

$\frac{1}{2} \log 3+\frac{20}{3}$

Question 50: Palette 50/82

AP-EAPCET 2024 MCQ

If $\int_1^n[x] d x=120$, then $n=$

A

15

B

16

C

14

D

12

Question 51: Palette 51/82

AP-EAPCET 2024 MCQ

$\int\limits_{\frac{-1}{24}}^{\frac{1}{24}} \sec x \log \left(\frac{1-x}{1+x}\right) d x=$

A

$\frac{\pi}{2}$

B

$\pi$

C

1

D

0

Question 52: Palette 52/82

AP-EAPCET 2024 MCQ

If $[x]$ is the greatest integer function, then $\int_0^5[x] d x=$

A

15

B

2

C

3

D

10

Question 53: Palette 53/82

AP-EAPCET 2024 MCQ

$\int_0^{\frac{\pi}{2}} \frac{1}{1+\sqrt{\tan x}} d x=$

A

0

B

$\frac{\pi}{2}$

C

$\frac{\pi}{3}$

D

$\frac{\pi}{4}$

Question 54: Palette 54/82

AP-EAPCET 2024 MCQ

$\int_0^\pi \frac{x \sin x}{1+\cos ^2 x} d x=$

A

0

B

$\frac{\pi}{2}$

C

$\frac{\pi^2}{2}$

D

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

Question 55: Palette 55/82

AP-EAPCET 2024 MCQ

$\int_{-\pi}^\pi \frac{x \sin ^3 x}{4-\cos ^2 x} d x=$

A

$2 \pi(1-\log 3)$

B

$2 \pi\left(1-\frac{3}{4} \log 3\right)$

C

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

D

$4 \pi(1-\log 3)$

Question 56: Palette 56/82

AP-EAPCET 2024 MCQ

$ \text { } \int\limits_{-3}^3|2-x| d x= $

A

12

B

16

C

13

D

25

Question 57: Palette 57/82

AP-EAPCET 2024 MCQ

$ \int_{\frac{1}{\sqrt[5]{31}}}^{\frac{1}{\sqrt[5]{242}}} \frac{1}{\sqrt[5]{x^{30}+x^{25}}} d x= $

A

$\frac{65}{4}$

B

$\frac{-75}{4}$

C

$\frac{75}{4}$

D

$\frac{-65}{4}$

Question 58: Palette 58/82

AP-EAPCET 2024 MCQ

If $\lim \limits_{n \rightarrow \infty}\left[\left(1+\frac{1}{n^2}\right)\left(1+\frac{4}{n^2}\right)\left(1+\frac{9}{n^2}\right) \ldots\left(1+\frac{n^2}{n^2}\right)\right]^{\frac{1}{n}}=a e^b$, then $ a+b= $

A

$\pi-2$

B

$\pi$

C

$\pi+2$

D

$\frac{\pi}{2}$

Question 59: Palette 59/82

AP-EAPCET 2024 MCQ

$ \int_0^\pi x \sin ^4 x \cos ^6 x d x= $

A

$\frac{3 \pi^2}{512}$

B

$\frac{3 \pi^2}{256}$

C

$\frac{\pi^2}{256}$

D

$\frac{\pi^2}{512}$

Question 60: Palette 60/82

AP-EAPCET 2024 MCQ

If $I_n=\int_0^{\frac{\pi}{4}} \tan ^n x d x$, then $I_{13}+I_{11}=$

A

$\frac{1}{13}$

B

$\frac{1}{12}$

C

$\frac{1}{10}$

D

$\frac{1}{11}$

Question 61: Palette 61/82

AP-EAPCET 2024 MCQ

$\lim \limits_{n \rightarrow+\infty}\left[{\frac{1}{n^4}+\frac{1}{\left(n^2+1\right)^{\frac{3}{2}}}+\frac{1}{\left(n^2+4\right)^{\frac{3}{2}}}+\frac{1}{\left(n^2+9\right)^{\frac{3}{2}}}}{+\ldots \ldots+\frac{1}{4 \sqrt{2} n^5}}\right]=$

A

$\frac{3}{4 \sqrt{2}}$

B

$\frac{3 \sqrt{2}}{4}$

C

$\frac{5}{6 \sqrt{2}}$

D

$\frac{5 \sqrt{2}}{6}$

Question 62: Palette 62/82

AP-EAPCET 2024 MCQ

$\int_{\log 4}^{\log 4} \frac{e^{2 x}+e^x}{e^{2 r}-5 e^x+6} d x=$

A

$\log \left(\frac{64}{9}\right)$

B

$\log \left(\frac{256}{81}\right)$

C

$\log \left(\frac{32}{3}\right)$

D

$\log \left(\frac{128}{27}\right)$

Question 63: Palette 63/82

AP-EAPCET 2024 MCQ

$\int_1^2 \frac{x^4-1}{x^6-1} d x=$

A

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

B

$\frac{121}{6}$

C

$\sqrt{2}-1$

D

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

Question 64: Palette 64/82

AP-EAPCET 2022 MCQ

Let $T>0$ be a fixed number. $f: R \rightarrow R$ is a continuous function such that $f(x+T)=f(x), x \in R$ If $I=\int_\limits0^T f(x) d x$, then $\int_\limits0^{5 T} f(2 x) d x=$

A

10 I

B

$\frac{5}{2} I$

C

5 I

D

2 I

Question 65: Palette 65/82

AP-EAPCET 2022 MCQ

$\int_\limits1^3 x^n \sqrt{x^2-1} d x=6 \text {, then } n=$

A

2

B

3

C

4

D

5

Question 66: Palette 66/82

AP-EAPCET 2022 MCQ

[ . ] represents greatest integer function, then $\int_{-1}^1(x[1+\sin \pi x]+1) d x=$

A

1

B

2

C

$\frac{5}{2}$

D

$\frac{3}{2}$

Question 67: Palette 67/82

AP-EAPCET 2022 MCQ

$\begin{aligned} & \lim _{n \rightarrow \infty}\left[\frac{n}{(n+1) \sqrt{2 n+1}}+\frac{n}{(n+2) \sqrt{2(2 n+2)}}\right. \\ & \left.+\frac{n}{(n+3) \sqrt{3(2 n+3)}}+\ldots n \text { terms }\right]=\int_\limits0^1 f(x) d x \end{aligned}$

then $f(x)=$

A

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

B

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

C

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

D

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

Question 68: Palette 68/82

AP-EAPCET 2022 MCQ

If $I_n=\int_0^{\pi / 4} \tan ^n x d x$, then $\frac{1}{I_2+I_4}+\frac{1}{I_3+I_5}+\frac{1}{I_4+I_6}=$

A

$\frac{1}{I_9+I_{11}}$

B

$\frac{1}{I_{10}+I_{12}}$

C

$\frac{1}{I_{12}+I_{14}}$

D

$\frac{1}{I_{11}+I_{13}}$

Question 69: Palette 69/82

AP-EAPCET 2022 MCQ

$\int_0^{\pi / 4} e^{\tan ^2 \theta} \sin ^2 \theta \tan \theta d \theta=$

A

$\frac{1}{2}\left(\frac{e}{2}-1\right)$

B

$\frac{e}{2}-1$

C

$\frac{\pi}{2}$

D

$2\left(\frac{\pi}{2}-e\right)$

Question 70: Palette 70/82

AP-EAPCET 2022 MCQ

$\int_{\pi / 4}^{5 \pi / 4}(|\cos t| \sin t+|\sin t| \cos t) d t=$

A

0

B

1

C

1/2

D

$\sqrt3/2$

Question 71: Palette 71/82

AP-EAPCET 2022 MCQ

If $f(x)=\max \{\sin x, \cos x\}$ and $g(x)=\min \{\sin x, \cos x\}$, then $\int_0^\pi f(x) d x+\int_0^\pi g(x) d x=$

A

$2 \sqrt{2}+2$

B

$2 \sqrt{2}-2$

C

2

D

$2 \sqrt{2}$

Question 72: Palette 72/82

AP-EAPCET 2022 MCQ

$\int_0^1 a^k x^k d x=$

A

$\lim _\limits{n \rightarrow \infty} \frac{a^k\left(1+2^k+3^k \ldots+n^k\right)}{n^{k+1}}$

B

$\lim _\limits{n \rightarrow \infty} \frac{a^k+a^k+\ldots+a^k}{n^{k+1}}$

C

$\lim _\limits{n \rightarrow \infty} \frac{1}{n} \Sigma\left(\frac{r}{n}\right)^k$

D

$\lim _\limits{n \rightarrow \infty} \frac{1}{n} \Sigma\left(\frac{2 r}{n}\right)^k$

Question 73: Palette 73/82

AP-EAPCET 2022 MCQ

Let $\alpha$ and $\beta(\alpha<\beta)$ are roots of $18 x^2-9 \pi x+\pi^2=0, f(x)=x^2, g(x)=\cos x$. Then, $\int_\alpha^\beta x(g \circ f(x)) d x=$

A

$\frac{\sqrt{3}-1}{4}$

B

$\frac{\sqrt{3}}{4}$

C

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

D

$\frac{1}{2}\left(\sin \frac{\pi^2}{9}-\sin \frac{\pi^2}{36}\right)$

Question 74: Palette 74/82

AP-EAPCET 2022 MCQ

$\int_0^\pi x\left(\sin ^2(\sin x)+\cos ^2(\cos x)\right) d x=$

A

$\pi^2$

B

$\pi^2 / 2$

C

$2 \pi$

D

$\pi / 4$

Question 75: Palette 75/82

AP-EAPCET 2021 MCQ

If $\int_\limits0^\pi \log (\sin x) d x=8 k$, then $\int_\limits0^{\frac{\pi}{4}} \log (1+\tan x) d x$ is equal to

A

$k$

B

$-k$

C

$\frac{k}{2}$

D

$4 k$

Question 76: Palette 76/82

AP-EAPCET 2021 MCQ

If $\int_\limits0^1 x^m(1-x)^n d x=k \int_\limits0^1 x^n(1-x)^m d x$, then the value of $k$ equals

A

$m$

B

$n$

C

$\frac{1}{\mathrm{mn}}$

D

$1$

Question 77: Palette 77/82

AP-EAPCET 2021 MCQ

If $\int_0^a {{{dx} \over {4 + {x^2}}} = {\pi \over 8}} $, then the value of a is equal to

A

1

B

2

C

3

D

4

Question 78: Palette 78/82

AP-EAPCET 2021 MCQ

$\int_1^2 {{{{x^3} - 1} \over {{x^2}}}} $ is equal to

A

${5 \over 3}$

B

${3 \over 5}$

C

1

D

$-$1

Question 79: Palette 79/82

AP-EAPCET 2021 MCQ

If $\int_0^{\pi / 2} \tan ^n(x) d x=k \int_0^{\pi / 2} \cot ^n(x) d x$, then

A

$k=1$

B

$k=2$

C

$k=\frac{1}{2}$

D

$k=3$

Question 80: Palette 80/82

AP-EAPCET 2021 MCQ

$\int_0^2 x e^x d x$ is equal to

A

$e^2+1$

B

$e^2-1$

C

$e^{-1}-1$

D

$e^{-1}+1$

Question 81: Palette 81/82

AP-EAPCET 2021 MCQ

$\int_2^4\{|x-2|+|x-3|\} d x$ is equal to

A

1

B

2

C

3

D

4

Question 82: Palette 82/82

AP-EAPCET 2021 MCQ

$\int\limits_{-1 / 2}^{1 / 2}\left\{[x]+\log \left(\frac{1+x}{1-x}\right)\right\} d x$ is equal to

A

$2 \log (1 / 2)$

B

0

C

$\frac{-1}{2}$

D

1