Question 1: Palette 1/69

TS-EAMCET 2025 MCQ

$ \lim _{n \rightarrow \infty} \frac{\left(2 n(2 n-1) \ldots .(n+2)(n+1)^{1 / n}\right.}{n}= $

A

$\int_0^1 \log x d x$

B

$\int_0^1 x \log x d x$

C

$\int_0^1(x+1) \log (x+1) d x$

D

$\int_0^1 \log (1+x) d x$

Question 2: Palette 2/69

TS-EAMCET 2025 MCQ

If $\int_0^{\frac{\pi}{2}} \tan ^{14}\left(\frac{x}{2}\right) d x=2\left[\sum_{n=1}^7 f(n)-\frac{\pi}{4}\right]$, then $f(n)=$

A

$\frac{(-1)^n}{n-1}$

B

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

C

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

D

$\frac{(-1)^{n+1}}{n+1}$

Question 3: Palette 3/69

TS-EAMCET 2025 MCQ

$ \int_{-4}^5 \frac{1}{\sqrt{20+x-x^2}} d x= $

A

$\frac{81 \pi}{8}$

B

$\frac{9 \pi}{2}$

C

$\pi$

D

$\frac{\pi}{10}$

Question 4: Palette 4/69

TS-EAMCET 2025 MCQ

$ \int_0^{\frac{\pi}{2}} \frac{d x}{\cos x-\sqrt{3} \sin x}= $

A

0

B

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

C

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

D

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

Question 5: Palette 5/69

TS-EAMCET 2025 MCQ

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

A

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

B

$\frac{\pi}{2}$

C

$\sqrt{2} \pi$

D

$2 \pi$

Question 6: Palette 6/69

TS-EAMCET 2025 MCQ

$ \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

$2 \pi \log 2$

Question 7: Palette 7/69

TS-EAMCET 2025 MCQ

$ \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

$\log 7$

Question 8: Palette 8/69

TS-EAMCET 2025 MCQ

$ \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

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

Question 9: Palette 9/69

TS-EAMCET 2025 MCQ

$ \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

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

Question 10: Palette 10/69

TS-EAMCET 2025 MCQ

$ \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

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

Question 11: Palette 11/69

TS-EAMCET 2025 MCQ

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

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

Question 12: Palette 12/69

TS-EAMCET 2025 MCQ

$ \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

$2 e^2+1$

Question 13: Palette 13/69

TS-EAMCET 2025 MCQ

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

Question 14: Palette 14/69

TS-EAMCET 2025 MCQ

$ \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

$\pi$

Question 15: Palette 15/69

TS-EAMCET 2025 MCQ

$ \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

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

Question 16: Palette 16/69

TS-EAMCET 2025 MCQ

$ \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

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

Question 17: Palette 17/69

TS-EAMCET 2024 MCQ

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

A

$\frac{\pi}{20}$

B

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

C

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

D

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

Question 18: Palette 18/69

TS-EAMCET 2024 MCQ

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

A

$\frac{1}{5}$

B

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

C

$\frac{1}{15}$

D

$\frac{1}{10}$

Question 19: Palette 19/69

TS-EAMCET 2024 MCQ

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

A

8

B

4

C

1

D

0

Question 20: Palette 20/69

TS-EAMCET 2024 MCQ

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

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

Question 21: Palette 21/69

TS-EAMCET 2024 MCQ

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

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

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

Question 22: Palette 22/69

TS-EAMCET 2024 MCQ

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

A

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

B

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

C

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

D

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

Question 23: Palette 23/69

TS-EAMCET 2024 MCQ

$ \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

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

Question 24: Palette 24/69

TS-EAMCET 2024 MCQ

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

A

$8+2 \log 2$

B

$8+\log 2$

C

$8+2 \log 5$

D

$4+\log 5$

Question 25: Palette 25/69

TS-EAMCET 2024 MCQ

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

A

$16 \sqrt{2}$

B

$32 \sqrt{2}$

C

$128 \sqrt{2}$

D

$64 \sqrt{2}$

Question 26: Palette 26/69

TS-EAMCET 2024 MCQ

If $f(x)=\int \frac{\sin 2 x+2 \cos x}{4 \sin ^2 x+5 \sin x+1} d x$ and $f(0)=0$, then $f(\pi / 6)=$

A

$\log \frac{3}{4}$

B

$2 \log 2$

C

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

D

1

Question 27: Palette 27/69

TS-EAMCET 2024 MCQ

$\int_{-2}^2 x^4\left(4-x^2\right)^{\frac{7}{2}} d x=$

A

$4 \pi$

B

$\frac{\pi}{16}$

C

$28 \pi$

D

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

Question 28: Palette 28/69

TS-EAMCET 2023 MCQ

$ \int_0^{\pi / 2} \frac{x \tan x \sec ^2 x}{\tan ^4 x+1} d x= $

A

$\pi^2 / 16$

B

$\pi^2 / 4$

C

$\pi^2 / 8$

D

$\pi^2 / 32$

Question 29: Palette 29/69

TS-EAMCET 2023 MCQ

$ \int_3^6 \frac{\sqrt{x}}{\sqrt{9-x}+\sqrt{x}} d x= $

A

$1 / 2$

B

$3 / 2$

C

2

D

1

Question 30: Palette 30/69

TS-EAMCET 2023 MCQ

$ \lim _{n \rightarrow \infty}\left[\left(1+\frac{1}{n^2}\right)\left(1+\frac{2^2}{n^2}\right) \ldots(2)\right]^{1 / n}= $

A

$2 e^{\pi-4}$

B

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

C

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

D

$\frac{1}{2} e^{\frac{\pi-4}{2}}$

Question 31: Palette 31/69

TS-EAMCET 2023 MCQ

$ \int_{-1}^1 x|x| d x= $

A

1

B

$1 / 2$

C

0

D

$2 / 3$

Question 32: Palette 32/69

TS-EAMCET 2023 MCQ

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

A

$2 / 3$

B

$3 / 10$

C

$4 / 15$

D

$5 / 18$

Question 33: Palette 33/69

TS-EAMCET 2023 MCQ

If $\int_0^3\left(3 x^2-4 x+2\right) d x=k$, then an integer root of $3 x^2-4 x+2=3 k / 5$ is

A

1

B

0

C

15

D

-1

Question 34: Palette 34/69

TS-EAMCET 2023 MCQ

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

A

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

B

1

C

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

D

$\frac{\pi}{4}$

Question 35: Palette 35/69

TS-EAMCET 2023 MCQ

If $[x]$ represents greatest integer function, then

$ \int_{-2}^2[2-x] d x= $

A

10

B

6

C

4

D

3

Question 36: Palette 36/69

TS-EAMCET 2023 MCQ

$ \int_0^2 \frac{x}{(2-x)^{\frac{3}{4}}} d x= $

A

$\frac{24}{5} 2^{\frac{1}{4}}$

B

$\frac{5}{24} 2^{\frac{3}{4}}$

C

$\frac{32}{5} 2^{\frac{1}{4}}$

D

$\frac{5}{12} 2^{\frac{3}{4}}$

Question 37: Palette 37/69

TS-EAMCET 2023 MCQ

$ \int_0^2 x^3(2-x)^4 d x= $

A

$\frac{128}{105}$

B

$\frac{16}{35}$

C

$\frac{256}{105}$

D

$\frac{32}{35}$

Question 38: Palette 38/69

TS-EAMCET 2023 MCQ

$ \int_0^3\left|x^2-3 x+2\right| d x= $

A

$\frac{11}{6}$

B

$\frac{5}{6}$

C

$\frac{3}{2}$

D

$\frac{2}{3}$

Question 39: Palette 39/69

TS-EAMCET 2023 MCQ

$ \int_{-\frac{\pi}{8092}}^{\frac{\pi}{8092}} \frac{\sec (2023 x)}{1+(2023)^{(2023 x)}} d x= $

A

$\frac{1}{2023 \sqrt{2}}+C$

B

$\frac{\log (\sqrt{2}+1)}{2023}+C$

C

$\frac{\log 2}{4046}+C$

D

$\frac{\sqrt{2}}{2023}+C$

Question 40: Palette 40/69

TS-EAMCET 2023 MCQ

$ \int_0^2 x^{\frac{5}{2}} \sqrt{2-x} d x= $

A

$\frac{5 \pi}{16}$

B

$\frac{5}{4}$

C

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

D

$\frac{5}{8}$

Question 41: Palette 41/69

TS-EAMCET 2023 MCQ

$ \int_0^{\pi / 4} \frac{\sec x}{1+2 \sin ^2 x} d x= $

A

$\frac{1}{3} \log (\sqrt{2}+1)+\frac{\pi \sqrt{2}}{12}$

B

$\frac{2}{3} \log (\sqrt{2}+1)+\frac{\pi \sqrt{2}}{6}$

C

$\frac{1}{6} \log (\sqrt{2}-1)+\frac{\pi}{12}$

D

$\frac{1}{4} \log (\sqrt{2}-1)-\frac{\pi \sqrt{3}}{6}$

Question 42: Palette 42/69

TS-EAMCET 2023 MCQ

$ \lim\limits_{n \rightarrow \infty}\left[\frac{1}{n^2} \sec ^2 \frac{1}{n^2}+\frac{2}{n^2} \sec ^2 \frac{4}{n^2}+\ldots \ldots+\frac{1}{n} \sec ^2 1\right]= $

A

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

B

$\frac{1}{2} \operatorname{cosec}(1)$

C

$\tan (1)$

D

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

Question 43: Palette 43/69

TS-EAMCET 2023 MCQ

$ \int\limits_2^5 \sqrt{\frac{5-x}{x-2}} d x= $

A

$\pi$

B

$\frac{\pi}{2}$

C

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

D

$\frac{\pi}{4}$

Question 44: Palette 44/69

TS-EAMCET 2023 MCQ

$ \int\limits_0^{\frac{\pi}{2}} \sin ^6 x \cos ^4 x d x= $

A

$\frac{\pi}{256}$

B

$\frac{\pi}{512}$

C

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

D

$\frac{5 \pi}{512}$

Question 45: Palette 45/69

TS-EAMCET 2023 MCQ

$ \int_{1 / 2}^2\left|\log _{10} x\right| d x= $

A

$\log _{10}\left(\frac{8}{e}\right)$

B

$\frac{1}{2} \log _{10}\left(\frac{8}{e}\right)$

C

$\log _{10}\left(\frac{2}{e}\right)$

D

$\log _e\left(\frac{3}{e}\right)$

Question 46: Palette 46/69

TS-EAMCET 2023 MCQ

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

A

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

B

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

C

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

D

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

Question 47: Palette 47/69

TS-EAMCET 2023 MCQ

[.] is the greatest integer function, then

$ \int_0^{2 \pi}[|\sin x|+|\cos x|] d x= $

A

$\frac{\pi}{2}$

B

$\pi$

C

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

D

$2 \pi$

Question 48: Palette 48/69

TS-EAMCET 2023 MCQ

If $f$ is defined on $R$ such that $f(x) f(-x)=9$, then $ \int_{-23}^{23} \frac{d x}{3+f(x)}= $

A

$\frac{51}{3}$

B

$\frac{49}{3}$

C

$\frac{46}{3}$

D

$\frac{46}{6}$

Question 49: Palette 49/69

TS-EAMCET 2022 MCQ

$ \int_0^4| | x-2|-x| d x= $

A

2

B

3

C

6

D

12

Question 50: Palette 50/69

TS-EAMCET 2022 MCQ

If $\int_{-a}^a f(x) d x=\int_0^a f(x) d x+\int_0^a g(x) d x$, then $g(x)=$

A

$-f(x)$

B

$f(x)$

C

$f(-x)$

D

$f(x)+f(-x)$.

Question 51: Palette 51/69

TS-EAMCET 2022 MCQ

76. Given that $\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{r=1}^{n p} f\left(\frac{r}{n}\right)=\int_0^p f(x) d x$. If $f: R \rightarrow R$ is defined by $f(x)=x^2+2$, then

$ \lim _{n \rightarrow \infty} \frac{3}{n}\left[f\left(\frac{7}{n}\right)+f\left(\frac{14}{n}\right)+f\left(\frac{21}{n}\right)+\ldots+f(7)\right]= $

A

55

B

57

C

104

D

7

Question 52: Palette 52/69

TS-EAMCET 2022 MCQ

If $f(x)=\left|\begin{array}{ccc}2 \cos ^2 x & \sin 2 x & \sin x \\ \sin 2 x & 2 \sin ^2 x & -\cos x \\ \sin x & -\cos x & 0\end{array}\right|$, then

$ \left.\int_0^{\pi / 4}|2| f(x) \mid+5 f^{\prime}(x)\right) d x= $

A

0

B

$\frac{\pi}{4}$

C

$\frac{\pi}{2}$

D

$\pi$

Question 53: Palette 53/69

TS-EAMCET 2022 MCQ

$\int_0^3\left(\sin \left(\frac{\pi}{3} x\right)-\cos \left(\frac{\pi}{3} x\right)\right) d x=$

A

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

B

0

C

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

D

$\frac{6}{\pi}$

Question 54: Palette 54/69

TS-EAMCET 2022 MCQ

$ \int_0^{\pi / 2} \sin ^4 \theta \cos ^3 \theta d \theta= $

A

$\frac{1}{35}$

B

$\frac{2}{35}$

C

$\frac{4}{35}$

D

$\frac{8}{35}$

Question 55: Palette 55/69

TS-EAMCET 2022 MCQ

It is given that $\frac{d}{d t}(t \log t-t)=\log t$, then $\exp \left(\int_0^1 2 x \log \left(1+x^2\right) d x\right)=$

A

$e$

B

2

C

$\frac{4}{e}$

D

$\frac{e}{4}$

Question 56: Palette 56/69

TS-EAMCET 2022 MCQ

$ \int_0^{2 a} f(x) d x= $

A

$2 \int_0^a f(x) d x$

B

$\int_0^a(f(x)+f(x+a)) d x$

C

0

D

$\int_0^{2 a} f(2 a+x) d x$

Question 57: Palette 57/69

TS-EAMCET 2022 MCQ

$ \int_1^2 x \sqrt{4-x^2} d x= $

A

$\sqrt{3}$

B

2

C

$1 / \sqrt{3}$

D

$1 / 2$

Question 58: Palette 58/69

TS-EAMCET 2022 MCQ

If $[x]$ denotes the greatest integer function of $x$ and

$ \int_{-3 / 2}^{3 / 2}[2 x-3] d x=k, \text { then }\left|k+\frac{1}{2}\right|= $

A

7

B

8

C

10

D

12

Question 59: Palette 59/69

TS-EAMCET 2022 MCQ

$ \int_1^4\left(x+\sqrt{x}+\frac{1}{x}\right) d x-\int_1^{2 \log 2} d x= $

A

$\frac{79}{6}$

B

$\frac{643}{6}$

C

$\frac{321}{5}$

D

64

Question 60: Palette 60/69

TS-EAMCET 2022 MCQ

Let $I=\int_{-\pi / 4}^{\pi / 4} \frac{1}{2-\cos 2 x}\left(\frac{\beta}{\pi}+\log \left(\frac{4+\sin x}{4-\sin x}\right)\right) d x$. Given that $\int \frac{d x}{1+k x^2}=\frac{1}{\sqrt{k}} \tan ^{-1}(\sqrt{k} x)+c, \tan ^{-1}(0)=0$ and $\tan ^{-1}(\sqrt{3})=\pi / 3$. Then, $3 I^2=$

A

4

B

9

C

16

D

1

Question 61: Palette 61/69

TS-EAMCET 2020 MCQ

If

$ f(x)=\left|\begin{array}{ccc} 1+\sin x+\sin 2 x+\sin 3 x & \frac{3+\sin 2 x}{2} & \frac{-2+\sin 3 x}{3} \\ 3+4 \sin x & \frac{3}{2} & \frac{4}{3} \sin x \\ 1+\sin x & \frac{1}{2} \sin x & \frac{1}{3} \end{array}\right| $

then $\int_0^{\pi / 2}\left(f(x)+f^{\prime}(x)\right) d x=$

A

$\frac{-1}{6}$

B

$\frac{-1}{9}$

C

$\frac{-2}{9}$

D

$\frac{1}{27}$

Question 62: Palette 62/69

TS-EAMCET 2020 MCQ

$ \lim _{n \rightarrow \infty} \frac{1}{n}\left[\frac{1}{n} \sin ^{-1} \frac{1}{n}+\frac{2}{n} \sin ^{-1} \frac{2}{n}+\ldots+\frac{\pi}{2}\right]= $

A

$\frac{\pi}{2}$

B

$\frac{\pi}{3}$

C

$\frac{\pi}{8}$

D

$\frac{\pi}{4}$

Question 63: Palette 63/69

TS-EAMCET 2020 MCQ

If $f(x)=\frac{1}{x^3} \int_5^x\left(2 u^2-u f^{\prime}(u) d u\right.$, then $f^{\prime}(5)=$

A

$\frac{13}{2}$

B

$\frac{2}{13}$

C

$\frac{13}{5}$

D

$\frac{5}{13}$

Question 64: Palette 64/69

TS-EAMCET 2020 MCQ

Assertion (A) $\int_{-a}^a f(x) d x=\int_0^a(f(x)+f(-x)) d x$

Reason (R) $\int_a^b f(x) d x=\int_{g(a)}^{g(b)} f(g(u)) g^{\prime}(u) d u$

The correct option among the following is

A

(A) is true, (R) is true and (R) is the correct explanation for (A)

B

(A) is true, (R) is true but (R) is not the correct explanation for (A)

C

(A) is true but (R) is false

D

(A) is false but (R) is true

Question 65: Palette 65/69

TS-EAMCET 2020 MCQ

If $\cos x+\cos 2 x+\ldots+\cos n x=\frac{A(x)}{2 \sin x / 2}$, then $\int_0^\pi A(x) d x=$

A

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

B

$\frac{-4 n}{2 n+1}$

C

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

D

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

Question 66: Palette 66/69

TS-EAMCET 2020 MCQ

$\mathop {\lim }\limits_{x \to \infty } \frac{\pi}{2 n}\left[\sin \frac{\pi}{2 n}+\sin \frac{2 \pi}{2 n}+\ldots+\sin \frac{\pi}{2}\right]= $

A

1

B

0

C

4

D

3

Question 67: Palette 67/69

TS-EAMCET 2020 MCQ

$ \int_0^{\pi / 2} \frac{d x}{4+5 \sin x} $

A

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

B

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

C

$2 \log 3$

D

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

Question 68: Palette 68/69

TS-EAMCET 2020 MCQ

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

A

e

B

$2 e$

C

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

D

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

Question 69: Palette 69/69

TS-EAMCET 2020 MCQ

$ \int_{\pi / 4}^{\pi / 2} \frac{3 d x}{1+e^{\sqrt{8} \sin \left(x-\frac{3 \pi}{8}\right)}}= $

A

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

B

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

C

$\frac{\pi}{8}$

D

$\frac{3}{8} \pi$