iCON Education - Indefinite Integration TS-EAMCET Test

Candidate Name

Student

Exam Name

TS-EAMCET

Test Name

Indefinite Integration

Subject

Mathematics

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Remaining Time

02:03:00

Question 1: Palette 1/123

TS-EAMCET 2025 MCQ

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

-1

Question 2: Palette 2/123

TS-EAMCET 2025 MCQ

$ \begin{aligned} &\text { If } y=f(x)^{g(x)} \text { and } \frac{d y}{d x}=y\left[H(x) f^{\prime}(x)+G(x) g^{\prime}(x)\right] \text {, then }\\ &\int \frac{G(x) H(x) f^{\prime}(x)}{g(x)} d x= \end{aligned} $

$\log (\log f(x))+C$

$\frac{[\log f(x)]^2}{2}+C$

$\frac{\log f(x)}{2}+C$

$x^2+C$

Question 3: Palette 3/123

TS-EAMCET 2025 MCQ

$ \begin{aligned} I_1 & =\int \frac{e^x}{e^{4 x}+e^{2 x}+1} d x, I_2 \\ & =\int \frac{e^{-x}}{e^{-4 x}+e^{-2 x}+1} d x, \text { then } I_2-I_1= \end{aligned} $

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

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

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

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

Question 4: Palette 4/123

TS-EAMCET 2025 MCQ

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

$\operatorname{sech}^{-1} \sqrt{x}$

$\operatorname{cosec}^{-1} \sqrt{x}$

$\log \left(\frac{1+x}{\sqrt{x}}\right)$

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

Question 5: Palette 5/123

TS-EAMCET 2025 MCQ

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

$1 / 2$

$3 / 4$

$3 / 8$

$1 / 8$

Question 6: Palette 6/123

TS-EAMCET 2025 MCQ

If $\int x^3 \sin 3 x d x=\frac{1}{27}[f(x) \cos 3 x+g(x) \sin 3 x]+C$, then $f(\mathrm{l})+g(\mathrm{l})=$

Question 7: Palette 7/123

TS-EAMCET 2025 MCQ

If $I_1=\int \sin ^6 x d x$ and $I_2=\int \cos ^6 x d x$, then $I_1+I_2=$

$\frac{5 x}{8}+\frac{3 \cos 4 x}{32}+C$

$\frac{1}{32}(20 x-3 \sin 4 x)+C$

$\frac{1}{32}(20 x+3 \sin 4 x)+C$

$\frac{5 x}{4}+\frac{3 \sin 4 x}{16}+C$

Question 8: Palette 8/123

TS-EAMCET 2025 MCQ

$ \int \frac{x+\cos x}{1-\sin x} d x= $

$x \tan \left(\frac{\pi}{4}+\frac{x}{2}\right)+C$

$x \tan \frac{x}{2}+C$

$x \cot \frac{x}{2}+C$

$x \cot \left(\frac{\pi}{4}+\frac{x}{2}\right)+C$

Question 9: Palette 9/123

TS-EAMCET 2025 MCQ

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

$-\frac{1}{2} \sinh ^{-1}\left(\frac{2-3 x}{\sqrt{7}(x+2)}\right)+C$

$-\frac{1}{2} \sin ^{-1}\left(\frac{2+3 x}{\sqrt{7}(x+2)}\right)+C$

$\frac{1}{2} \cosh ^{-1}\left(\frac{2+3 x}{\sqrt{7}(x+2)}\right)+C$

$-\frac{1}{2} \cos ^{-1}\left(\frac{2-3 x}{\sqrt{7}(x+2)}\right)+C$

Question 10: Palette 10/123

TS-EAMCET 2025 MCQ

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

-1

Question 11: Palette 11/123

TS-EAMCET 2025 MCQ

$ \int \frac{3^x(x \log 3-1)}{x^2} d x= $

$x \cdot 3^x+C$

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

$x^2 3^x+C$

$\frac{3^x}{x}+C$

Question 12: Palette 12/123

TS-EAMCET 2025 MCQ

If $\frac{5 \pi}{4} < x < \frac{7 \pi}{4}$, then $\int \sqrt{\frac{1-\sin 2 x}{1+\sin 2 x}} d x=$

$-\sec ^2\left(\frac{\pi}{4}-x\right)+C$

$-\log \left|\sec \left(\frac{\pi}{4}-x\right)\right|+C$

$\sec ^2\left(\frac{\pi}{4}-x\right)+C$

$\log \left|\sec \left(\frac{\pi}{4}-x\right)\right|+C$

Question 13: Palette 13/123

TS-EAMCET 2025 MCQ

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

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

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

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

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

Question 14: Palette 14/123

TS-EAMCET 2025 MCQ

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

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

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

$\frac{\cos x}{\cos x+2 \sin x}+C$

$-\frac{\cos x}{2 \cos x+\sin x}+C$

Question 15: Palette 15/123

TS-EAMCET 2025 MCQ

If $\frac{x+3}{(x+1)\left(x^2+2\right)}=\frac{a}{x+1}+\frac{b x+c}{x^2+2}$, then $a-b+c=$

Question 16: Palette 16/123

TS-EAMCET 2025 MCQ

$ \int e^{-x}\left(x^3-2 x^2+3 x-4\right) d x= $

$-e^{-x}\left(x^3-x^2+5 x-1\right)+C$

$e^{-x}\left(x^3-x^2+5 x-1\right)+C$

$e^{-x}\left(x^3+x^2+5 x+1\right)+C$

$-e^{-x}\left(x^3+x^2+5 x+1\right)+C$

Question 17: Palette 17/123

TS-EAMCET 2025 MCQ

$ \int\left(1+\tan ^2 x\right)(1+2 x \tan x) d x= $

$x \sec x+C$

$x \tan ^2 x+C$

$x \sec ^2 x+C$

$x \tan x+C$

Question 18: Palette 18/123

TS-EAMCET 2025 MCQ

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

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

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

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

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

Question 19: Palette 19/123

TS-EAMCET 2025 MCQ

$ \int \frac{\log x}{(1+x)^3} d x= $

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

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

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

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

Question 20: Palette 20/123

TS-EAMCET 2025 MCQ

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

$\frac{8}{5}$

$\frac{16}{5}$

$\frac{3}{5}$

$\frac{19}{5}$

Question 21: Palette 21/123

TS-EAMCET 2025 MCQ

$ \int\left(\frac{1}{x^2}+\frac{\sin ^3 x+\cos ^3 x}{\sin ^2 x \cos ^2 x}\right) d x= $

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

$-\frac{1}{x}+\frac{\sin x+\cos x}{\cos x-\sin x}+c$

$-\frac{1}{x}+\frac{\sin x-\cos x}{\sin ^2 x \cos ^2 x}+C$

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

Question 22: Palette 22/123

TS-EAMCET 2025 MCQ

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

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

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

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

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

Question 23: Palette 23/123

TS-EAMCET 2025 MCQ

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

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

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

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

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

Question 24: Palette 24/123

TS-EAMCET 2025 MCQ

If $\int \frac{d x}{\left(x^2+9\right) \sqrt{x^2+16}}=\frac{1}{3 \sqrt{7}} \tan ^{-1}\left(K \frac{x}{\sqrt{16+x^2}}\right)+c$, then $K=$

$\frac{\sqrt{7}}{3}$

$3 \sqrt{7}$

$\frac{3}{\sqrt{7}}$

$\frac{3}{7}$

Question 25: Palette 25/123

TS-EAMCET 2025 MCQ

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

$\frac{1}{25}[17 \log |4 \cos x-3 \sin x|-6 x]+C$

$\frac{1}{25}[x-18 \log |4 \cos x-3 \sin x|]+C$

$\frac{1}{25}[\log |4 \cos x-3 \sin x|-18 x]+C$

$\frac{1}{25}[17 x-6 \log |4 \cos x-3 \sin x|]+C$

Question 26: Palette 26/123

TS-EAMCET 2025 MCQ

$ \int e^{4 x}(\sin 3 x-\cos 3 x) d x= $

$\frac{e^{4 x}}{25}(7 \sin 3 x-\cos 3 x)+C$

$\frac{e^{4 x}}{25}(\sin 3 x-7 \cos 3 x)+C$

$\frac{e^{4 x}}{5}(7 \sin 3 x+\cos 3 x)+C$

$\frac{e^{4 x}}{5}(\sin 3 x+7 \cos 3 x)+C$

Question 27: Palette 27/123

TS-EAMCET 2025 MCQ

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

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

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

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

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

Question 28: Palette 28/123

TS-EAMCET 2025 MCQ

If $\int(x+2) \sqrt{x^2-x+2} d x=\frac{1}{3} f(x)+\frac{5}{8} g(x)+\frac{35}{16} h(x)+C$ then $f(-1)+g(-1)+h\left(\frac{1}{2}\right)=$

-4

$2+\ln \left(\frac{\sqrt{7}}{2}\right)$

-2

Question 29: Palette 29/123

TS-EAMCET 2024 MCQ

$\int \frac{\sec x}{3(\sec x+\tan x)+2} d x=$

$\frac{1}{2} \log \left|\frac{\tan \frac{x}{2}+1}{\tan \frac{x}{2}+5}\right|+C$

$\frac{2}{\sqrt{11}} \tan ^{-1}\left\{\frac{3 \tan \frac{x}{2}+4}{\sqrt{11}}\right\}+C$

$\log |3 \sec x+2 \tan x|+C$

$\log |3 \tan x+2 \sec x|+C$.

Question 30: Palette 30/123

TS-EAMCET 2024 MCQ

$\int \frac{d x}{4+3 \cot x} d x=$

$-\frac{3}{25} \log |4+3 \cot x|+\frac{4}{25} x+c$

$-\frac{3}{25} \log |4 \sin x+3 \cos x|+\frac{4}{25} x+c$

$\frac{4}{25} \log |4 \sin x+3 \cos x|-\frac{3}{25} x+c$

$\frac{4}{25} \log |4+3 \cot x|-\frac{3}{25} x+c$

Question 31: Palette 31/123

TS-EAMCET 2024 MCQ

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

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

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

$-\frac{1}{\sqrt{5}} \sinh ^{-1}\left(\frac{4-x}{2(x+1)}\right)+c$

$-\frac{1}{\sqrt{5}} \cosh ^{-1}\left(\frac{4+x}{2(x-1)}\right)+c$

Question 32: Palette 32/123

TS-EAMCET 2024 MCQ

If $\int e^{x}\left(x^{3}+x^{2}-x+4\right) d x=e^{x} f(x)+c$, then $f(1)=$

Question 33: Palette 33/123

TS-EAMCET 2024 MCQ

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

$a$

$2 a$

$3 a$

$4 a$

Question 34: Palette 34/123

TS-EAMCET 2024 MCQ

If $\int \frac{1}{x^{4}+8 x^{2}+9} d x=\frac{1}{k}$\left[\frac{1}{\sqrt{14}} \tan ^{-1}(f(x))-\frac{1}{\sqrt{2}} \tan ^{-1}(g(x))\right]+c$ then,

$\sqrt{\frac{k}{2}+f(\sqrt{3})+g(1)}=$

$3-2 \sqrt{2}$

$\sqrt{2}-1$

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

$\sqrt{2}+1$

Question 35: Palette 35/123

TS-EAMCET 2024 MCQ

If $\int\left(1+x-x^{-1}\right) e^{\left(x+x^{-1}\right)} d x=f(x)+C$, then $f(1)-f(-1)=$

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

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

$e+\frac{1}{e}$

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

Question 36: Palette 36/123

TS-EAMCET 2024 MCQ

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

$\frac{1}{m-1}\left(\frac{\sqrt[m]{x^{m}+1}}{x}\right)^{m}+C$

$\frac{-1}{m-1}\left(\frac{\sqrt[m]{x^{m}+1}}{x}\right)^{m-1}+C$

$\frac{-1}{m}\left(\frac{\sqrt[m]{x^{m}+1}}{x}\right)^{m}+C$

$\frac{1}{m}\left(\frac{\sqrt[m-1]{x^{m}+1}}{x}\right)^{m}+C$

Question 37: Palette 37/123

TS-EAMCET 2024 MCQ

If $\int(\sqrt{\operatorname{cosec} x+1}) d x=k \tan ^{-1}(f(x))+C$, then $\frac{1}{k} f\left(\frac{\pi}{6}\right)=$

$\frac{1}{2}$

$\frac{1}{4}$

$-\frac{1}{4}$

$-\frac{1}{2}$

Question 38: Palette 38/123

TS-EAMCET 2024 MCQ

$\int \sqrt{4 \cos ^2 x-5 \sin ^2 x} \cos x d x=$

$\frac{1}{2} \cos x \sqrt{4-9 \sin ^2 x}+\frac{2}{3} \sin ^{-1}\left(\frac{3 \sin x}{2}\right)+c$

$\frac{1}{2} \sin x \sqrt{4-9 \sin ^2 x}+\frac{2}{3} \cos ^{-1}\left(\frac{3 \cos x}{2}\right)+c$

$\frac{1}{2} \cos x \sqrt{1-9 \cos ^2 x}+\frac{2}{3} \sin ^{-1}\left(\frac{3 \cos x}{2}\right)+c$

$\frac{1}{2} \sin x \sqrt{4-9 \sin ^2 x}+\frac{2}{3} \sin ^{-1}\left(\frac{3 \sin x}{2}\right)+c$

Question 39: Palette 39/123

TS-EAMCET 2024 MCQ

$\int\left(\frac{4 \tan ^4 x+3 \tan ^2 x-1}{\tan ^2 x+4}\right) d x=$

$4 \tan x-\frac{17}{4} \tan ^{-1}\left(\frac{\tan x}{4}\right)+c$

$4 \tan x-\frac{17}{4} \tan ^{-1}\left(\frac{\tan x}{2}\right)+c$

$4 \tan x-\frac{17}{2} \tan ^{-1}\left(\frac{\tan x}{2}\right)+c$

$2 \tan x-\frac{17}{2} \tan ^{-1}\left(\frac{\tan x}{2}\right)+c$

Question 40: Palette 40/123

TS-EAMCET 2024 MCQ

$\int\left(\frac{\left(\sin ^4 x+2 \cos ^2 x-1\right) \cos x}{(1+\sin x)^6}\right) d x=$

$\frac{\sin ^6 x}{6(1+\sin x)^6}+c$

$-\frac{\sin ^6 x}{6(1+\sin x)^6}+c$

$\frac{\cos ^6 x}{6(1+\sin x)^6}+c$

$-\frac{\cos ^6 x}{6(1+\sin x)^6}+c$

Question 41: Palette 41/123

TS-EAMCET 2024 MCQ

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

$(\log x)^3-3(\log x)^2+6 \log x-6+c$

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

$(x \log x)^3-3 x(\log x)^2+6 x(\log x)-6+c$

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

Question 42: Palette 42/123

TS-EAMCET 2024 MCQ

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

-11

-39

Question 43: Palette 43/123

TS-EAMCET 2024 MCQ

$\int e^{-2 x}\left(\tan 2 x-2 \sec ^2 2 x \tan 2 x\right) d x=$

$e^{-2 x} \tan 2 x+c$

$-\frac{e^{-2 x}}{2}\left[\sec ^2 2 x+\tan 2 x\right]+c$

$-\frac{e^{-2 x}}{2}\left[\tan 2 x-\sec ^2 2 x\right]+c$

$e^{-2 x} \sec ^2 2 x+c$

Question 44: Palette 44/123

TS-EAMCET 2024 MCQ

If $\int x^3 \sin 3 x d x=f(x) \cos 3 x+g(x) \sin 3 x+C$, then 27 $(f(x)+x g(x))=$

$18 x^3+4 x$

$8 x$

$4 x$

$18 x^3+8 x$

Question 45: Palette 45/123

TS-EAMCET 2024 MCQ

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

$\frac{1}{25} \tan ^{-1}\left(\frac{3}{4} \sec ^2 2 x\right)+c$

$\frac{1}{25} \tan ^{-1}\left(\frac{4}{3} \sec ^2 2 x\right)+c$

$\frac{1}{24} \tan ^{-1}\left(\frac{3}{4} \tan 2 x\right)+c$

$\frac{1}{24} \tan ^{-1}\left(\frac{4}{3} \tan 2 x\right)+c$

Question 46: Palette 46/123

TS-EAMCET 2024 MCQ

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

$\frac{7}{15} \log |\cos 3 x+2 \sin 3 x|-\frac{4}{5} x+c$

$-\frac{4}{5} \log |\cos 3 x+2 \sin 3 x|+\frac{7 x}{5}+c$

$\frac{7}{5} \log |\cos 3 x+2 \sin 3 x|-\frac{4}{5} x+c$

$-\frac{8}{15} \log |\cos 3 x+2 \sin 3 x|+\frac{x}{5}+c$

Question 47: Palette 47/123

TS-EAMCET 2024 MCQ

If $\frac{x^4}{\left(x^2+1\right)(x-2)}=f(x)+\frac{A x+B}{x^2+1}+\frac{C}{x-2}$, then $f(14)+2 A-B=$

$5 C$

$4 C$

$6 C$

$7 C$

Question 48: Palette 48/123

TS-EAMCET 2024 MCQ

$\int(\sqrt{1-\sin x}+\sqrt{1+\sin x}) d x=f(x)+c$, where $c$ is the constant of integration. If $\frac{5 \pi}{2}$<$x<\frac{7 \pi}{2}$ and $ f\left(\frac{8 \pi}{3}\right)=-2, \text { then } f^{\prime}\left(\frac{8 \pi}{3}\right)= $

$\sqrt{3}$

-1

Question 49: Palette 49/123

TS-EAMCET 2024 MCQ

$\int \frac{\left(1-4 \sin ^2 x\right) \cos x}{\cos (3 x+2)} d x=$

$(\cos 2) x-\frac{1}{3}(\sin 2) \log |\sec (3 x+2)|+c$

$(\sin 2) x-\frac{1}{3}(\cos 2) \log |\cos (3 x+2)|+c$

$(\sin 2) x+\frac{1}{3}(\cos 2) \log |\cos (3 x+2)|+c$

$(\cos 2) x+\frac{1}{3}(\sin 2) \log |\sec (3 x+2)|+c$

Question 50: Palette 50/123

TS-EAMCET 2024 MCQ

$\int \frac{\left(1-4 \sin ^2 x\right) \cos x}{\cos (3 x+2)} d x=$

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

$\log \left(x^5+x^2\right)-\log \left(x^3+x\right)+\log (x+1)+c$

$\frac{2}{9} x^8-\frac{4}{9} x^6+\frac{1}{9} x^4-\frac{1}{3} x^2+c$

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

Question 51: Palette 51/123

TS-EAMCET 2024 MCQ

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

$\frac{1}{m-1}\left(\frac{\sqrt[m]{x^{m}+1}}{x}\right)^{m}+C$

$\frac{-1}{m-1}\left(\frac{\sqrt[m]{x^{m}+1}}{x}\right)^{m-1}+C$

$\frac{-1}{m}\left(\frac{\sqrt[m]{x^{m}+1}}{x}\right)^{m}+C$

$\frac{1}{m}\left(\frac{\sqrt[m-1]{x^{m}+1}}{x}\right)^{m}+C$

Question 52: Palette 52/123

TS-EAMCET 2023 MCQ

$ \int \frac{\tan x}{\sec ^2 x\left(1+\sec ^6 x\right)^{\frac{2}{3}}} d x= $

$\frac{-1}{2}\left(1+\sec ^6 x\right)^{\frac{1}{3}}+C$

$2\left(1+\sec ^6 x\right)^{\frac{4}{3}}+C$

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

$2\left(1+\cos ^6 x\right)^{\frac{1}{3}}+C$

Question 53: Palette 53/123

TS-EAMCET 2023 MCQ

$ \int \frac{1}{(x-1)^{\frac{5}{7}}(x+1)^{\frac{9}{7}}} d x= $

$\frac{7}{4}\left(\frac{x+1}{x-1}\right)^{\frac{2}{7}}+C$

$\frac{-7}{4}\left(\frac{x+1}{x-1}\right)^{\frac{2}{7}}+C$

$\frac{7}{4}\left(\frac{x-1}{x+1}\right)^{\frac{2}{7}}+C$

$\frac{-7}{4}\left(\frac{x-1}{x+1}\right)^{\frac{2}{7}}+C$

Question 54: Palette 54/123

TS-EAMCET 2023 MCQ

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

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

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

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

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

Question 55: Palette 55/123

TS-EAMCET 2023 MCQ

$ \begin{aligned} & \text { If } \int \frac{1}{\operatorname{cosec} x+\cos x} d x=\frac{1}{2 \sqrt{3}} \log |f(x)| \\ & -\int \frac{\cos x-\sin x}{2+\sin 2 x} d x+c, \text { then at } x=\frac{\pi}{3},|f(x)|= \end{aligned} $

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

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

$\frac{6 \sqrt{3}-2}{\sqrt{3}+1}$

$\frac{6 \sqrt{3}+2}{\sqrt{3}+1}$

Question 56: Palette 56/123

TS-EAMCET 2023 MCQ

If $\int \frac{x^{49} \tan ^{-1}\left(x^{50}\right)}{\left(1+x^{100}\right)} d x=k\left(\tan ^{-1}\left(x^{50}\right)\right)^2+C$, then $k=$

$\frac{-1}{100}$

$\frac{1}{50}$

$\frac{-1}{50}$

$\frac{1}{100}$

Question 57: Palette 57/123

TS-EAMCET 2023 MCQ

If $\int(\log x)^3 x^5 d x=\frac{x^6}{A}\left[B(\log x)^3\right. \left.+C(\log x)^2+D(\log x)-1\right]+k$ and $A, B, C, D$ are integers, then $A-(B+C+D)=$

172

184

192

216

Question 58: Palette 58/123

TS-EAMCET 2023 MCQ

$ \int \frac{d x}{\left(x^2+1\right)\left(x^2+4\right)}= $

$\frac{1}{3} \tan ^{-1} x+\frac{1}{6} \tan ^{-1}\left(\frac{x}{2}\right)+C$

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

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

$\frac{1}{3} \tan ^{-1} x-\frac{1}{6} \tan ^{-1}\left(\frac{x}{2}\right)+C$

Question 59: Palette 59/123

TS-EAMCET 2023 MCQ

$\int \frac{d x}{(x-1)^{\frac{3}{4}}(x+2)^{\frac{5}{4}}}=$

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

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

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

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

Question 60: Palette 60/123

TS-EAMCET 2023 MCQ

$\int \frac{1}{\left(x+\frac{2}{x}\right) \sqrt{x^4+4 x^2+3}} d x=$

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

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

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

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

Question 61: Palette 61/123

TS-EAMCET 2023 MCQ

If $\frac{3 \pi}{2} < x < \frac{5 \pi}{2}$ and $\int(\sqrt{1-\sin x}+\sqrt{1+\sin x}) d x=f(x)+C$, where $C$ is the constant of integration, then $f\left(\frac{\pi}{3}\right)-f(0)=$

-2

$2 \sqrt{2}$

$-2 \sqrt{2}$

Question 62: Palette 62/123

TS-EAMCET 2023 MCQ

If $\int \frac{2 \sin 2 x-3 \cos x}{2 \sin ^2 x-3 \sin x+4} d x=f(x)+C$, where $C$ is the constant of integration, then $f\left(\frac{\pi}{2}\right)-f(0)=$

$2 \log 2$

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

Question 63: Palette 63/123

TS-EAMCET 2023 MCQ

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

$\frac{2}{3} \sqrt{3 x^2-2 x+1}+\frac{11}{3} \sinh ^{-1}\left(\frac{3 x-1}{\sqrt{2}}\right)+C$

$\frac{1}{3} \sqrt{3 x^2-2 x+1}+\frac{11}{3} \sinh ^{-1}\left(\frac{\sqrt{3} x-1}{\sqrt{2}}\right)+C$

$\frac{1}{3} \sqrt{3 x^2-2 x+1}+\frac{11}{3} \sinh ^{-1}\left(\frac{3 x-1}{\sqrt{3}}\right)+C$

$\frac{2}{3} \sqrt{3 x^2-2 x+1}+\frac{11}{3 \sqrt{3}} \sinh ^{-1}\left(\frac{3 x-1}{\sqrt{2}}\right)+C$

Question 64: Palette 64/123

TS-EAMCET 2023 MCQ

$ \int \frac{1}{16-7 \sin ^2 x} d x= $

$\frac{1}{12} \tan ^{-1}\left(\frac{3 \tan x}{4}\right)+C$

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

$\frac{1}{12} \log \left(\frac{4-\sqrt{7} \sin x}{4+\sqrt{7} \sin x}\right)+C$

$\frac{1}{12} \log \left(\frac{4+\sqrt{7} \sin x}{4-\sqrt{7} \sin x}\right)+C$

Question 65: Palette 65/123

TS-EAMCET 2023 MCQ

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

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

$-\frac{1+3(\sec x+\tan x)^2}{6(\sec x+\tan x)^3}+C$

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

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

Question 66: Palette 66/123

TS-EAMCET 2023 MCQ

$ \int \frac{1}{3 \cos x-4 \sin x+5} d x= $

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

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

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

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

Question 67: Palette 67/123

TS-EAMCET 2023 MCQ

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

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

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

$\frac{1}{5}\left[\log \frac{|x-2|}{\sqrt{7+x^2}}+2 \tan ^{-1} x\right]+C$

$\frac{1}{5}\left[\log \frac{|x-2|}{\sqrt{1+x^2}}-2 \tan ^{-1} x\right]+C$

Question 68: Palette 68/123

TS-EAMCET 2023 MCQ

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

$-\frac{1}{2}$

$\frac{1}{2}$

$\frac{3}{2}$

Question 69: Palette 69/123

TS-EAMCET 2023 MCQ

$ \text { Match the following items from List I into List II } $

List-I	List-II
1. $\int \frac{\sin^{2} x}{\cos^{4} x} d x$	A. $\frac{\tan^{2} x}{2} + \ln \| \cos x \| + C$
2. $\int \frac{\sin^{4} x}{\cos^{2} x} d x$	B. $\cos x + \sec x + C$
3. $\int \frac{\sin^{3} x}{\cos^{2} x} d x$	C. $\frac{\tan^{3} x}{3} + C$
4. $\int \frac{\sin^{3} x}{\cos^{3} x} d x$	D. $\tan x + \frac{\sin 2 x}{4} - \frac{3 x}{2} + C$
	E. $\cos x - \sec x + C$

Select the correct choice

1-C, 2-E, 3-B, 4-A

1,-C, 2-D, 3-B, 4-A

1-D, 2-C, 3-A, 4-B

1-C, 2-E, 3-A, 4-D

Question 70: Palette 70/123

TS-EAMCET 2023 MCQ

If $\int \frac{x}{(a+x)^5} d x=\frac{1}{k(a+x)^4}(f(x))+C$, then $\frac{f(-a)}{a k}=$

$1 / 3$

$1 / 2$

$5 / 6$

$1 / 4$

Question 71: Palette 71/123

TS-EAMCET 2023 MCQ

If $\int x^4(\log x)^3 d x=x^5\left[A(\log x)^3\right]$ $\left.+B(\log x)^2+C \log x+D\right]+k$, then $A+B+C+5 D=$

$\frac{2}{25}$

$\frac{8}{25}$

$\frac{12}{125}$

$\frac{16}{125}$

Question 72: Palette 72/123

TS-EAMCET 2023 MCQ

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

$-17 / 2$

Question 73: Palette 73/123

TS-EAMCET 2023 MCQ

$ \int \frac{\sin ^8 x-\cos ^8 x}{1-2 \sin ^2 x \cos ^2 x} d x= $

$\frac{1}{2} \cos 2 x+c$

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

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

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

Question 74: Palette 74/123

TS-EAMCET 2023 MCQ

If $\int \frac{x^2\left(\sec ^2 x+\tan x\right)}{(x \tan x+1)^2} d x=\frac{-x^2}{x \tan x+1}+f(x)+C$, then $ f(x)= $

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

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

$2 \log |x \sin x+\cos x|+C$

$2 \log |x \cos x+\sin x|+C$

Question 75: Palette 75/123

TS-EAMCET 2023 MCQ

7. If $\int \sin (101 x)(\sin x)^{99} d x$ $=\frac{\sin (100 x)(\sin x)^\lambda}{\mu}+C$ then, $\frac{\lambda}{\mu}=$

Question 76: Palette 76/123

TS-EAMCET 2023 MCQ

If $\int e^x\left(\sin ^2 2 x-8 \cos 4 x\right) d x=e^x f(x)+C$, then $f\left(\frac{\pi}{4}\right)=$

-1

$e$

Question 77: Palette 77/123

TS-EAMCET 2023 MCQ

If $n$ is a positive integer greater than 1 and $I_n=\int \frac{\sin n x}{\sin x} d x$, then $I_{n+1}-I_{n-1}=$

$\frac{2}{n-1} \cos (n-1) x$

$\frac{2}{n-1} \sin (n-1) x$

$\frac{2}{n} \cos n x$

$\frac{2}{n} \sin n x$

Question 78: Palette 78/123

TS-EAMCET 2022 MCQ

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

$3 B$

$2 B$

$B$

Question 79: Palette 79/123

TS-EAMCET 2022 MCQ

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

-1

Question 80: Palette 80/123

TS-EAMCET 2022 MCQ

If $f(x)=\int \frac{2-3 \sin ^2 x}{1+\cos 2 x} d x$ and $f\left(\frac{\pi}{4}\right)=1$, then $f(0)=$

$\frac{3}{8}(4-\pi)$

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

Question 81: Palette 81/123

TS-EAMCET 2022 MCQ

If $x \neq(2 n+1) \frac{\pi}{2}, n \in Z$ and $\cos x \neq \frac{-1}{2}$, then

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

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

$\frac{\sec ^3 x}{3}-x+c$

$\cot x-x+c$

$\tan x-x+c$

Question 82: Palette 82/123

TS-EAMCET 2022 MCQ

Given that $\int \frac{1}{x^2+a^2} d x=\frac{1}{a} \tan ^{-1}\left(\frac{x}{a}\right)+c$.

$ \begin{aligned} & \text { If } \int \frac{1}{x^4+3 x^2+1} d x=a \tan ^{-1}\left(\frac{b\left(x^2-1\right)}{x}\right) \\ & +c \tan ^{-1}\left(\frac{d\left(x^2+1\right)}{x}\right)+k \end{aligned} $

where $k$ is a constant of integration, then $5(c+d+a b)=$

Question 83: Palette 83/123

TS-EAMCET 2022 MCQ

If $\frac{2 x^2-3 x+5}{(x-7)^3}=\frac{A}{x-7}+\frac{B}{(x-7)^2}+\frac{C}{(x-7)^3}$, then $2 A-3 B+C=$

Question 84: Palette 84/123

TS-EAMCET 2022 MCQ

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

-2

-3

Question 85: Palette 85/123

TS-EAMCET 2022 MCQ

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

$\frac{2^x}{\log 2}-\sin x+\cos x-\frac{1}{x}-\log x+c$

$2^x \log 2+\sin x-\cos x-\frac{1}{x}+\frac{1}{x^2}+c$

$\frac{2^x}{\log 2}+\sin x-\cos x-\frac{1}{x}-\log x+c$

$2^x \log 2-\sin x+\cos x-\frac{1}{x}+\frac{1}{x^2}+c$

Question 86: Palette 86/123

TS-EAMCET 2022 MCQ

If $\tan \alpha=\frac{4}{3}$, then $\int \frac{1}{3 \cos x-4 \sin x} d x=$

$\frac{1}{5} \log \left|\tan \left(\frac{x}{2}+\frac{\alpha}{2}\right)\right|+c$

$\frac{1}{5} \log \left|\tan \left(\frac{\pi}{4}+\frac{x}{2}+\frac{\alpha}{2}\right)\right|+c$

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

$\frac{1}{5} \log |\tan (\sec x+\tan x)|+c$

Question 87: Palette 87/123

TS-EAMCET 2022 MCQ

If $x \neq(2 n+1) \frac{\pi}{2}$, then $\int \frac{\cos ^3 x}{(1+\sin x)^4} d x=$

$-\frac{\cos ^4 x}{(1+\sin x)^3}+c$

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

$-\frac{\cos ^4 x}{(1+\sin x)^4}+c$

$-\frac{\cos ^4 x}{4(1+\sin x)^4}+c$

Question 88: Palette 88/123

TS-EAMCET 2022 MCQ

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

$\frac{-3}{2}$

$\frac{-1}{2}$

$\frac{5}{2}$

Question 89: Palette 89/123

TS-EAMCET 2022 MCQ

Let $g(x)$ be the anti-derivative of $f(x)$. Then, the function for which $\log _e\left(1+(g(x))^2\right)+c$ is an anti-derivative is

$\left(1+(g(x))^2\right) g^{\prime}(x) f(x)$

$\frac{-2 f(x) g(x)}{1+g(x)}$

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

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

Question 90: Palette 90/123

TS-EAMCET 2022 MCQ

If $f(x)=\int\left[\tan ^2 x+\cot ^2 x+\frac{4\left(\sin ^3 x+\cos ^3 x\right)}{\sin ^2 2 x}\right] d x$ and $f\left(\frac{\pi}{4}\right)=0$, then $3\left[f\left(\frac{\pi}{6}\right)+2\right]=$

$\frac{\pi}{2}$

$\frac{\pi}{4}$

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

Question 91: Palette 91/123

TS-EAMCET 2022 MCQ

$\int \sqrt{4 \cos ^2 x-5 \sin ^2 x} \cos x d x=$

$\frac{1}{2} \sin x \sqrt{4-9 \sin ^2 x}+\frac{2}{3} \sin ^{-1}\left(\frac{3 \sin x}{2}\right)+c$

$\frac{1}{2} \cos x \sqrt{4-9 \cos ^2 x}+\frac{2}{3} \sin ^{-1}\left(\frac{3 \cos x}{2}\right)+c$

$\frac{1}{2} \sin x \sqrt{4-9 \sin ^2 x}+\frac{2}{3} \cos ^{-1}\left(\frac{3 \cos x}{2}\right)+c$

$\frac{1}{2} \cos x \sqrt{4-9 \sin ^2 x}+\frac{2}{3} \sin ^{-1}\left(\frac{3 \sin x}{2}\right)+c$

Question 92: Palette 92/123

TS-EAMCET 2022 MCQ

If $\frac{42-13 x}{x^2+x-6}=\frac{A}{l x+m}+\frac{B}{p x+q}$, where $l m>0$ and $p q<0$, then $\frac{A l p}{B m q}=$

$\frac{27}{32}$

$\frac{27}{8}$

$\frac{8}{243}$

$\frac{243}{32}$

Question 93: Palette 93/123

TS-EAMCET 2022 MCQ

Given that $\frac{d}{d x}\left(\tan ^{-1} x\right)=\frac{1}{1+x^2}$ and $\frac{d}{d x}\left(\sin h^{-1} x\right)=\frac{1}{\sqrt{1+x^2}}$. Then, $\int \frac{3 x^6-2 x^4+x^2-2}{x^2+1} d x=$

$\frac{3}{7} x^7-\frac{2}{5} x^5+\frac{1}{3} x^3-2 x+c$

$\frac{\frac{3}{7} x^7-\frac{2}{5} x^5+\frac{1}{3} x^3-2 x}{\frac{x^3}{3}+x}+c$

$\frac{3}{5} x^5-\frac{5}{3} x^3+6 x-8 \tan ^{-1} x+c$

$\frac{3}{5} x^5-\frac{5}{3} x^3+6 x-8 \sin h^{-1} x+c$

Question 94: Palette 94/123

TS-EAMCET 2022 MCQ

$ \int \frac{\sin x \cdot \sec ^2 x-\tan x \cdot \sin x+\cos x}{(1-\cos 2 x)} d x= $

$\frac{1}{2}\left[\sec x-\operatorname{cosec} x-\log \left|\tan \left(\frac{x}{2}\right) \tan \left(\frac{\pi}{4}+\frac{x}{2}\right)\right|\right]+C$

$\sec x-\operatorname{cosec} x+\log \left|\frac{\tan \left(\frac{\pi}{2}\right)}{\tan \left(\frac{\pi}{4}+\frac{x}{2}\right)}\right|+C$

$\frac{1}{2}\left[\sec x-\operatorname{cosec} x-\log \left|\frac{\tan \left(\frac{\pi}{4}+\frac{x}{2}\right)}{\tan \left(\frac{x}{2}\right)}\right|\right]+C$

$\sec x+\operatorname{cosec} x-\log \left|\tan \left(\frac{x}{2}\right)\right|+C$

Question 95: Palette 95/123

TS-EAMCET 2022 MCQ

If $f(x)=\int \frac{16 x^7+5 x^{10}}{\left(x^3+2+3 x^8\right)^2} d x(x \geq 0)$ and $f(0)=1$, then the value of $f(-1)$ is

$\frac{7}{6}$

$\frac{5}{4}$

$\frac{-3}{4}$

$\frac{-5}{6}$

Question 96: Palette 96/123

TS-EAMCET 2022 MCQ

$ \begin{aligned} & \text { If } \int \frac{(x+3)}{(x-1)^2(2 x-1)} d x \\ & =\frac{A}{x-1}+B \log (2 x-1)+C \log (x-1)+k, \text { then } A+B+C= \end{aligned} $

-4

-11

Question 97: Palette 97/123

TS-EAMCET 2022 MCQ

If $\int \frac{1+\cos 8 x}{\tan 2 x-\cot 2 x} d x=f(x) \cdot \cos (g(x))+c$, then $f\left(\frac{1}{4}\right)+g\left(\frac{1}{4}\right)=$

$\frac{17}{8}$

$\frac{15}{8}$

$\frac{33}{16}$

Question 98: Palette 98/123

TS-EAMCET 2022 MCQ

Let $x \neq \frac{-3}{5}, \frac{2}{5}$, if $f\left(\frac{2 x+1}{5 x+3}\right)=x+2$, then $\int f(x) d x=$

$\frac{7}{5} x-\frac{1}{5} \log |5 x+3|+c$

$\frac{7}{5} x-\frac{1}{25} \log |5 x+3|+c$

$\frac{7}{5} x-\frac{1}{25} \log |5 x-2|+c$

$\frac{7}{5} x-\frac{1}{5} \log |5 x-2|+c$

Question 99: Palette 99/123

TS-EAMCET 2022 MCQ

If $\int e^x \cos x d x=\frac{e^x}{2}(\cos x+\sin x)$ and

$ \int \frac{\cos \left(\log \left(\frac{2 x+3}{3-2 x}\right)\right)}{(3-2 x)^2} d x=\frac{f(x)}{24}[\cos (g(x))+\sin (g(x))]+c $

then $g(1)=$

$\log f(2)$

$\log f(1)$

Question 100: Palette 100/123

TS-EAMCET 2022 MCQ

If $\frac{x^2-3 x+2}{(x-4)(x-3)^2}=\frac{A}{x-4}+\frac{B}{x-3}+\frac{C}{(x-3)^2}$ then $A+B+C=$

-1

Question 101: Palette 101/123

TS-EAMCET 2022 MCQ

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

Question 102: Palette 102/123

TS-EAMCET 2022 MCQ

Let $f(x)=\int \frac{2 x^3-3 x^2+4 x-5}{x^2} d x$ and $f(1)=1$. Then, $f(5)=$

$10+4 \log 5$

$10-4 \log 5$

$9+4 \log 5$

$9-4 \log 5$

Question 103: Palette 103/123

TS-EAMCET 2022 MCQ

If $x>0$ and $x \neq(2 n+1) \frac{\pi}{2}$, then $\int\left(x \sqrt{x}-e^{\log (\sec x \tan x)}+\frac{3 x^2-2 x+1}{x^2}\right) d x=$

$x \sqrt{x}-\sec x+3 x-2 \log x-\frac{1}{x}+c$

$\frac{2}{5} x^2 \sqrt{x}-\sec x+3 x+\frac{2}{x^2}-\frac{1}{x}+c$

$x \sqrt{x}-\sec x+3 x+\frac{2}{x^2}-\frac{1}{x}+c$

$\frac{2}{5} x^2 \sqrt{x}-\sec x+3 x-2 \log x-\frac{1}{x}+c$

Question 104: Palette 104/123

TS-EAMCET 2022 MCQ

$ \int(2 x-3) \sqrt{3 x+2} d x= $

$\frac{2}{135}\left(54 x^2-123 x+106\right) \sqrt{3 x+2}+c$

$\frac{2}{135}\left(54 x^2+123 x-106\right) \sqrt{3 x+2}+c$

$\frac{2}{135}\left(54 x^2-123 x-106\right) \sqrt{3 x+2}+c$

$\frac{2}{135}\left(54 x^2-195 x-106\right) \sqrt{3 x+2}+c$

Question 105: Palette 105/123

TS-EAMCET 2020 MCQ

If the partial fractions decomposition of $\frac{x^4+24 x^2+28}{\left(x^2+1\right)^3}$ is $\frac{A}{x^2+1}+\frac{B}{\left(x^2+1\right)^2}+\frac{C}{\left(x^2+1\right)^3}$ then $B-2 A+C=$

Question 106: Palette 106/123

TS-EAMCET 2020 MCQ

$ \int \frac{x^2}{\left(\sqrt{4-x^2}\right)^3} d x= $

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

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

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

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

Question 107: Palette 107/123

TS-EAMCET 2020 MCQ

$ \int \frac{d x}{x \ln (x) \ln ^2(x) \ln ^3(x) \ldots \ln ^m(x)}=\frac{(\ln (x))^K}{K}+C \Rightarrow 2 K= $

$(m+1)(m+2)$

$(2-m)(1-m)$

$(m+1)(2-m)$

$(m+2)(1-m)$

Question 108: Palette 108/123

TS-EAMCET 2020 MCQ

If $I_m=\int x^m \cos n x d x=g(x)-\frac{m(m-1)}{n^2} I_{m-2}$, then $g(x)=$

$\frac{x^m \sin n x}{n}+\frac{m(m-1) x^{m-1} \cos n x}{n^2}$

$\frac{x^m \cos n x}{n}+\frac{x^{m-1} m(m-1)}{n^2} \sin n x$

$\frac{m}{n} \sin n x+\frac{m}{n^2} x^{m-1} \cos n x$

$\frac{x^m \sin n x}{n}+\frac{m}{n^2} x^{m-1} \cos n x$

Question 109: Palette 109/123

TS-EAMCET 2020 MCQ

Let $I_n=\int \sec ^n x d x$. If $5 I_6-4 I_4=f(x)$, then $f\left(\frac{\pi}{4}\right)$ is equal to

$4 / 5$

Question 110: Palette 110/123

TS-EAMCET 2020 MCQ

If $\int \frac{(x-1) d x}{(x+1) \sqrt{x^3+x^2+x}}=A \cdot \tan ^{-1} \sqrt{f(x)}+$ constant, then the ordered pair $(A, f(-1))=$

$(2,1)$

$(2,-1)$

$(1,2)$

$(-2,2)$

Question 111: Palette 111/123

TS-EAMCET 2020 MCQ

If $f\left(\frac{2 x+3}{3 x+5}\right)=x+4, x \neq \frac{-5}{3}, \frac{2}{3}$ and $\int f(x) d x=A x+B \ln |3 x-2|+C$, then $3 B-A=$

$\frac{64}{9}$

$\frac{-52}{21}$

$\frac{-10}{3}$

$\frac{-8}{3}$

Question 112: Palette 112/123

TS-EAMCET 2020 MCQ

If $\int e^x\left(\frac{x^2-8 x+19}{(x-1)^5}\right) d x=\frac{e^x(l x+m)}{(x-1)^4}+C$, then $4 l+m=$

-5

-2

Question 113: Palette 113/123

TS-EAMCET 2020 MCQ

$ \int \frac{d x}{(x-2) \sqrt{x^2-3 x+5}}= $

$\frac{-1}{\sqrt{3}} \cosh ^{-1}\left[\frac{7 x-8}{\sqrt{37}(x-2)}\right]+C$

$\frac{-1}{\sqrt{3}} \sinh ^{-1}\left[\frac{x+4}{\sqrt{11}(x-2)}\right]+C$

$\frac{-1}{\sqrt{3}} \cosh ^{-1}\left[\frac{x+4}{\sqrt{11}(x-2)}\right]+C$

$\frac{-1}{\sqrt{3}} \sinh ^{-1}\left[\frac{7 x-8}{\sqrt{37}(x-2)}\right]+C$

Question 114: Palette 114/123

TS-EAMCET 2020 MCQ

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

$\frac{3}{4}$

$\frac{1}{2}$

Question 115: Palette 115/123

TS-EAMCET 2020 MCQ

For $x \in\left(\frac{3 \pi}{4}, \pi\right), \int(\sqrt{1+\sin 2 x}+\sqrt{1-\sin 2 x}) d x=$

$-2 \cos x+C$

$2 \sin x+C$

$-2 \sin x+C$

$2 \cos x+C$

Question 116: Palette 116/123

TS-EAMCET 2020 MCQ

$ \begin{aligned} & \text { If } \int \frac{x^2\left(x \sec ^2 x+\tan x\right)}{(x \tan x+1)^2} d x=A \log (|x \sin x+\cos x|) \\ & +B \frac{f(x)}{(x \tan x+1)}+C \text {, then } f(A+B)= \end{aligned} $

-1

Question 117: Palette 117/123

TS-EAMCET 2020 MCQ

$ \text { If } \begin{aligned} & \int x^3(\log x)^2 d x=x^4\left[A(\log x)^2+B(\log x)\right. \\ &+C \log e]+K, \text { then } A+B+C \end{aligned} $

$\frac{7}{24}$

$\frac{4}{25}$

$\frac{3}{14}$

$\frac{5}{32}$

Question 118: Palette 118/123

TS-EAMCET 2020 MCQ

$ \begin{aligned} & \text { If } \int \frac{9 x+15}{x^3-6 x-9} d x=A \log |g(x)| \\ & \quad+B \log |f(x)|+C, \text { then } \frac{(A-B) g(4)}{f(-1)}= \end{aligned} $

$\frac{1}{7}$

$\frac{3}{7}$

Question 119: Palette 119/123

TS-EAMCET 2020 MCQ

If $\frac{4 x^2+5 x^4+7}{\left(x^2+1\right)\left(x^4+x^2+1\right)}=\frac{A x+B}{x^2+1} +\frac{C x^3+D x^2+E x+F}{x^4+x^2+1}$, then $B+2(D+F+E)-C \cdot A=$

-3

Question 120: Palette 120/123

TS-EAMCET 2020 MCQ

$ \int \frac{y^2+\sqrt[3]{y^4}+\sqrt[6]{y^2}}{y\left(1+\sqrt[3]{y^2}\right)} d y= $

$\frac{3}{4} \sqrt[3]{y^4}+3 \tan ^{-1}(\sqrt[3]{y})+C$

$\frac{3}{2} y^{2 / 3}+6 \tan ^{-1}\left(\sqrt[6]{y^2}\right)+C$

$\frac{2}{3 \sqrt[3]{y^2}}+6 \log \left(1+y^2\right)+C$

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

Question 121: Palette 121/123

TS-EAMCET 2020 MCQ

For $k \in(1, \infty), \int \frac{1}{1+k \cos x} d x=$

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

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

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

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

Question 122: Palette 122/123

TS-EAMCET 2020 MCQ

$ \int e^{-3 x}\left(x^2+\sin 4 x\right) d x= $

$-e^{-3 x}\left(\frac{x^2}{3}+\frac{2 x}{9}+\frac{2}{27}+\frac{3}{25} \sin 4 x+\frac{4}{25} \cos 4 x\right)+C$

$-e^{-3 x}\left(\frac{x^2}{3}-\frac{2 x}{9}+\frac{2}{27}+\frac{3}{25} \sin 4 x+\frac{4}{25} \cos 4 x\right)+C$

$-e^{-3 x}\left(\frac{x^2}{3}+\frac{2 x}{9}+\frac{2}{27}+\frac{3}{25} \sin 4 x-\frac{4}{25} \cos 4 x\right)+C$

$-e^{-3 x}\left(\frac{x^2}{3}-\frac{2 x}{9}+\frac{2}{27}+\frac{3}{25} \sin 4 x-\frac{4}{25} \cos 4 x\right)+C$

Question 123: Palette 123/123

TS-EAMCET 2020 MCQ

If $\int \frac{2 x^{12}+5 x^9}{\left(1+x^3+x^5\right)^3} d x=\frac{x^m}{l\left(1+x^3+x^5\right)^r}+C$, then $\frac{m-l}{r}=$

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