Definite Integration - Calculus - Mathematics Previous Year Questions

1

2024 · Mathematics · Calculus · Definite Integration

TG EAPCET 2024 ONLINE 10TH MAY EVENING SHIFT

If $m, l, r, s, n$ are integers such that $9 > m > l > s > n > r > 2$ and $\int_{-2 \pi}^{2 \pi} \sin ^{m} x \cos ^{n} x d x=4 \int_{0}^{\pi} \sin ^{m} x \cos ^{n} x d x, \int_{-\pi}^{\pi} \sin ^{r} x \cos ^{s} x d x$ $=4 \int_{0}^{\pi / 2} \sin ^{r} x \cos ^{s} x d x$ and $\int_{-\pi / 2}^{\pi / 2} \sin ^{l} x \cos ^{m} x d x=0$, then

A

$(s-2)(1-2)=m r$

B

$(s-2)(l+2)=r m+5$

C

$(s-2)(s+2)=\ln -3$

D

$(I-2)(I+2)=m s-5$

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2

2024 · Mathematics · Calculus · Definite Integration

TG EAPCET 2024 ONLINE 10TH MAY EVENING SHIFT

$\frac{3}{25} \int_{0}^{25 \pi} \sqrt{\left|\cos x-\cos ^{3} x\right|} d x=$

A

8

B

4

C

1

D

0

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3

2024 · Mathematics · Calculus · Definite Integration

TG EAPCET 2024 ONLINE 10TH MAY MORNING SHIFT

$\int_0^\pi\left(\sin ^3 x+\cos ^2 x\right)^2 d x=$

A

$\frac{15 \pi}{16}+\frac{8}{15}$

B

$\frac{11 \pi}{16}+\frac{8}{15}$

C

$\frac{15 \pi}{16}+\frac{4}{15}$

D

$\frac{11 \pi}{16}+\frac{4}{15}$

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4

2024 · Mathematics · Calculus · Definite Integration

TG EAPCET 2024 ONLINE 10TH MAY MORNING SHIFT

$\int_{\frac{-\pi}{8}}^{\frac{\pi}{8}} \frac{\sin ^4(4 x)}{1+e^{4 x}} d x=$

A

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

B

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

C

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

D

$\frac{3 \pi}{32}$

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5

2024 · Mathematics · Calculus · Definite Integration

TG EAPCET 2024 ONLINE 11TH MAY MORNING SHIFT

$\int_{-\pi / 15}^{\pi / 5} \frac{\cos 5 x}{1+e^{5 x}} d x=$

A

$\frac{1}{5}$

B

$\frac{\sqrt{3}}{10}$

C

$\frac{1}{15}$

D

$\frac{1}{10}$

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6

2024 · Mathematics · Calculus · Definite Integration

TG EAPCET 2024 ONLINE 11TH MAY MORNING SHIFT

$\int_{\frac{\pi}{5}}^{\frac{3 \pi}{10}} \frac{d x}{\sec ^{2} x+\left(\tan ^{2024} x-1\right)\left(\sec ^{2} x-1\right)}=$

A

$\frac{\pi}{20}$

B

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

C

$\frac{3 \pi}{20}$

D

$\frac{3 \pi}{5}$

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7

2024 · Mathematics · Calculus · Definite Integration

TG EAPCET 2024 ONLINE 9TH MAY EVENING SHIFT

$\int_0^{32 \pi} \sqrt{1-\cos 4 x} d x=$

A

$16 \sqrt{2}$

B

$32 \sqrt{2}$

C

$128 \sqrt{2}$

D

$64 \sqrt{2}$

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8

2024 · Mathematics · Calculus · Definite Integration

TG EAPCET 2024 ONLINE 9TH MAY EVENING SHIFT

$ \int_{\frac{-3}{4}}^{\frac{\pi-6}{8}} \log (\sin (4 x+3)) d x= $

A

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

B

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

C

$-\frac{\pi}{14} \log 2$

D

$-\frac{\pi}{28} \log 2$

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9

2024 · Mathematics · Calculus · Definite Integration

TG EAPCET 2024 ONLINE 9TH MAY EVENING SHIFT

$\int_0^{16} \frac{\sqrt{x}}{1+\sqrt{x}} d x=$

A

$8+2 \log 2$

B

$8+\log 2$

C

$8+2 \log 5$

D

$4+\log 5$

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10

2025 · Mathematics · Calculus · Definite Integration

TG EAPCET 2025 ONLINE 2ND MAY EVENING SHIFT

Let $m, n, p, q$ be four positive integers. If

$$\begin{aligned} & \int_0^{2 \pi} \sin ^m x \cos ^n x d x=4 \int_0^{\pi / 2} \sin ^m x \cos ^n x d x \int_0^{2 \pi} \sin ^p x \cos ^n x d x=0 \\ & \int_0^\pi \sin ^p x \cos ^q x d x=0, a=m+n+p \text { and } b=m+n+q, \text { then } \end{aligned}$$

A

$a$ is even number and $b$ is odd number

B

$a$ is odd number and $b$ is even number

C

Both $a$ and $b$ are even numbers

D

Both $a$ and $b$ are odd numbers

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11

2025 · Mathematics · Calculus · Definite Integration

TG EAPCET 2025 ONLINE 2ND MAY EVENING SHIFT

$$\mathop {\lim }\limits_{n \to \infty } \frac{1}{n^2}\left[e^{1 / n}+2 e^{2 / n}+3 e^{3 / n}+\ldots+2 n e^2\right]=$$

A

$e^2-1$

B

$e^2+1$

C

$2 e^2-2$

D

$2 e^2+1$

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12

2025 · Mathematics · Calculus · Definite Integration

TG EAPCET 2025 ONLINE 2ND MAY MORNING SHIFT

$$\int_0^{\pi / 4} x^2 \sin 2 x d x$$

A

$\frac{\pi^2-2}{8}$

B

$\frac{\pi(\pi-2)}{8}$

C

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

D

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

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13

2025 · Mathematics · Calculus · Definite Integration

TG EAPCET 2025 ONLINE 2ND MAY MORNING SHIFT

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

A

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

B

$\frac{9 \pi}{32}$

C

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

D

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

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14

2025 · Mathematics · Calculus · Definite Integration

TG EAPCET 2025 ONLINE 2ND MAY MORNING SHIFT

$$\int_0^2 \sqrt{(x+3)(2-x)} d x=$$

A

$\frac{25}{8} \cos ^{-1}\left(\frac{1}{5}\right)-\frac{\sqrt{6}}{4}$

B

$\frac{25}{8} \sin ^{-1}\left(\frac{1}{5}\right)-\frac{\sqrt{6}}{4}$

C

$\frac{\pi}{2}$

D

$\pi$

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15

2025 · Mathematics · Calculus · Definite Integration

TG EAPCET 2025 ONLINE 3RD MAY EVENING SHIFT

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

A

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

B

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

C

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

D

$2 \pi \log 2$

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16

2025 · Mathematics · Calculus · Definite Integration

TG EAPCET 2025 ONLINE 3RD MAY EVENING SHIFT

$$\int_{-2}^4\left|2-x^2\right| d x=$$

A

$\frac{8 \sqrt{2}}{3}-3$

B

$\frac{16 \sqrt{2}}{3}+12$

C

$\frac{16 \sqrt{2}}{3}-3$

D

$\frac{8 \sqrt{2}}{3}+12$

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17

2025 · Mathematics · Calculus · Definite Integration

TG EAPCET 2025 ONLINE 3RD MAY EVENING SHIFT

$$\int_0^{\frac{\pi}{4}} \frac{\sec x}{3 \cos x+4 \sin x} d x=$$

A

$\log \left(\frac{7}{3}\right)$

B

$\frac{1}{4} \log \left(\frac{7}{3}\right)$

C

$\frac{1}{4} \log 7$

D

$\log 7$

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18

2025 · Mathematics · Calculus · Definite Integration

TG EAPCET 2025 ONLINE 3RD MAY MORNING SHIFT

If [.] denotes the greatest integer function, then $\int_1^2\left[x^2\right] d x=$

A

$5+\sqrt{2}+\sqrt{3}$

B

$5+\sqrt{2}-\sqrt{3}$

C

$5-\sqrt{2}-\sqrt{3}$

D

$5-\sqrt{2}+\sqrt{3}$

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19

2025 · Mathematics · Calculus · Definite Integration

TG EAPCET 2025 ONLINE 3RD MAY MORNING SHIFT

$$\int_8^{18} \frac{1}{(x+2) \sqrt{x-3}} d x=$$

A

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

B

$\frac{\pi}{6}$

C

$\frac{\pi}{3}$

D

$\frac{\pi}{3 \sqrt{5}}$

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20

2025 · Mathematics · Calculus · Definite Integration

TG EAPCET 2025 ONLINE 3RD MAY MORNING SHIFT

$$\int_0^{\pi / 4} \frac{1}{5 \cos ^2 x+16 \sin ^2 x+8 \sin x \cos x} d x=$$

A

$\tan ^{-1}\left(\frac{4}{5}\right)$

B

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

C

$\frac{1}{8} \tan ^{-1}\left(\frac{8}{9}\right)$

D

$\frac{1}{4} \tan ^{-1}\left(\frac{7}{8}\right)$

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