Indefinite Integration - Calculus - Mathematics Previous Year Questions

1

2024 · Mathematics · Calculus · Indefinite Integration

TG EAPCET 2024 ONLINE 10TH MAY EVENING SHIFT

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,

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

A

$3-2 \sqrt{2}$

B

$\sqrt{2}-1$

C

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

D

$\sqrt{2}+1$

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2

2024 · Mathematics · Calculus · Indefinite Integration

TG EAPCET 2024 ONLINE 10TH MAY EVENING SHIFT

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

A

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

B

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

C

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

D

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

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3

2024 · Mathematics · Calculus · Indefinite Integration

TG EAPCET 2024 ONLINE 10TH MAY EVENING SHIFT

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

A

$\frac{1}{2}$

B

$\frac{1}{4}$

C

$-\frac{1}{4}$

D

$-\frac{1}{2}$

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4

2024 · Mathematics · Calculus · Indefinite Integration

TG EAPCET 2024 ONLINE 10TH MAY EVENING SHIFT

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

A

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

B

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

C

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

D

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

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5

2024 · Mathematics · Calculus · Indefinite Integration

TG EAPCET 2024 ONLINE 10TH MAY EVENING SHIFT

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

$a$

B

$2 a$

C

$3 a$

D

$4 a$

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6

2024 · Mathematics · Calculus · Indefinite Integration

TG EAPCET 2024 ONLINE 10TH MAY MORNING SHIFT

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

A

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

B

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

C

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

D

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

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7

2024 · Mathematics · Calculus · Indefinite Integration

TG EAPCET 2024 ONLINE 10TH MAY MORNING SHIFT

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

A

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

B

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

C

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

D

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

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8

2024 · Mathematics · Calculus · Indefinite Integration

TG EAPCET 2024 ONLINE 10TH MAY MORNING SHIFT

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

A

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

B

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

C

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

D

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

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9

2024 · Mathematics · Calculus · Indefinite Integration

TG EAPCET 2024 ONLINE 10TH MAY MORNING SHIFT

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

A

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

B

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

C

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

D

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

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10

2024 · Mathematics · Calculus · Indefinite Integration

TG EAPCET 2024 ONLINE 11TH MAY MORNING SHIFT

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

A

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

B

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

C

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

D

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

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11

2024 · Mathematics · Calculus · Indefinite Integration

TG EAPCET 2024 ONLINE 11TH MAY MORNING SHIFT

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

A

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

B

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

C

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

D

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

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12

2024 · Mathematics · Calculus · Indefinite Integration

TG EAPCET 2024 ONLINE 11TH MAY MORNING SHIFT

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

A

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

B

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

C

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

D

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

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13

2024 · Mathematics · Calculus · Indefinite Integration

TG EAPCET 2024 ONLINE 11TH MAY MORNING SHIFT

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

A

0

B

1

C

2

D

3

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14

2024 · Mathematics · Calculus · Indefinite Integration

TG EAPCET 2024 ONLINE 9TH MAY EVENING SHIFT

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

A

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

B

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

C

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

D

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

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15

2024 · Mathematics · Calculus · Indefinite Integration

TG EAPCET 2024 ONLINE 9TH MAY EVENING SHIFT

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

A

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

B

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

C

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

D

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

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16

2024 · Mathematics · Calculus · Indefinite Integration

TG EAPCET 2024 ONLINE 9TH MAY EVENING SHIFT

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

A

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

B

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

C

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

D

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

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17

2024 · Mathematics · Calculus · Indefinite Integration

TG EAPCET 2024 ONLINE 9TH MAY EVENING SHIFT

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

A

$18 x^3+4 x$

B

$8 x$

C

$4 x$

D

$18 x^3+8 x$

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18

2024 · Mathematics · Calculus · Indefinite Integration

TG EAPCET 2024 ONLINE 9TH MAY EVENING SHIFT

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 $2 A+3 B-C-D+E=$

A

0

B

1

C

-11

D

-39

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19

2025 · Mathematics · Calculus · Indefinite Integration

TG EAPCET 2025 ONLINE 2ND MAY EVENING SHIFT

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

A

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

B

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

C

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

D

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

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20

2025 · Mathematics · Calculus · Indefinite Integration

TG EAPCET 2025 ONLINE 2ND MAY EVENING SHIFT

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 $K=$

A

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

B

$3 \sqrt{7}$

C

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

D

$\frac{3}{7}$

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