Definite Integration · Mathematics · AP EAPCET

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MCQ (Single Correct Answer)

1

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

AP EAPCET 2024 - 23th May Morning Shift
2
$\int_0^\pi \frac{x \sin x}{4 \cos ^2 x+3 \sin ^2 x} d x$ is equal to
AP EAPCET 2024 - 23th May Morning Shift
3
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
AP EAPCET 2024 - 23th May Morning Shift
4
$\int_0^1 \sqrt{\frac{2+x}{2-x}} d x$ is equal to
AP EAPCET 2024 - 22th May Evening Shift
5
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
AP EAPCET 2024 - 22th May Evening Shift
6
$\int\limits_{-2}^2\left(4-x^2\right)^{\frac{5}{2}} d x$ is equal to
AP EAPCET 2024 - 22th May Evening Shift
7

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

AP EAPCET 2024 - 22th May Evening Shift
8
$\int\limits_{-5 \pi}^{5 \pi}(1-\cos 2 x)^{\frac{5}{2}} d x$ is equal to
AP EAPCET 2024 - 22th May Evening Shift
9

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

AP EAPCET 2024 - 22th May Morning Shift
10

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

AP EAPCET 2024 - 22th May Morning Shift
11
$$\int\limits_0^{\pi /4} {{{{x^2}} \over {{{(x\,\sin \,x + \cos \,x)}^2}}}dx = } $$
AP EAPCET 2024 - 21th May Evening Shift
12
$\int_0^1 \frac{x}{(1-x)^{\frac{3}{4}}} d x=$
AP EAPCET 2024 - 21th May Evening Shift
13

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

AP EAPCET 2024 - 21th May Evening Shift
14
$\int_1^5(|x-3|+|1-x|) d x=$
AP EAPCET 2024 - 21th May Evening Shift
15
If $729 \int_1^3 \frac{1}{x^3\left(x^2+9\right)^2} d x=a+\log b$, then $(a-b)=$
AP EAPCET 2024 - 21th May Morning Shift
16
$\lim \limits_{n \rightarrow \infty} \frac{1^{17}+2^{77}+\ldots+n^{77}}{n^{78}}=$
AP EAPCET 2024 - 21th May Morning Shift
17

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

AP EAPCET 2024 - 21th May Morning Shift
18
If $\int_1^n[x] d x=120$, then $n=$
AP EAPCET 2024 - 21th May Morning Shift
19
$\int\limits_{\frac{-1}{24}}^{\frac{1}{24}} \sec x \log \left(\frac{1-x}{1+x}\right) d x=$
AP EAPCET 2024 - 20th May Evening Shift
20
If $[x]$ is the greatest integer function, then $\int_0^5[x] d x=$
AP EAPCET 2024 - 20th May Evening Shift
21
$\int_0^{\frac{\pi}{2}} \frac{1}{1+\sqrt{\tan x}} d x=$
AP EAPCET 2024 - 20th May Evening Shift
22
$\int_0^\pi \frac{x \sin x}{1+\cos ^2 x} d x=$
AP EAPCET 2024 - 20th May Evening Shift
23
$\int_{-\pi}^\pi \frac{x \sin ^3 x}{4-\cos ^2 x} d x=$
AP EAPCET 2024 - 20th May Morning Shift
24

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

AP EAPCET 2024 - 20th May Morning Shift
25

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

AP EAPCET 2024 - 20th May Morning Shift
26
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 $$ a+b= $$
AP EAPCET 2024 - 19th May Evening Shift
27
$$ \int_0^\pi x \sin ^4 x \cos ^6 x d x= $$
AP EAPCET 2024 - 19th May Evening Shift
28
If $I_n=\int_0^{\frac{\pi}{4}} \tan ^n x d x$, then $I_{13}+I_{11}=$
AP EAPCET 2024 - 19th May Evening Shift
29
$\lim \limits_{n \rightarrow+\infty}\left[{\frac{1}{n^4}+\frac{1}{\left(n^2+1\right)^{\frac{3}{2}}}+\frac{1}{\left(n^2+4\right)^{\frac{3}{2}}}+\frac{1}{\left(n^2+9\right)^{\frac{3}{2}}}}{+\ldots \ldots+\frac{1}{4 \sqrt{2} n^5}}\right]=$
AP EAPCET 2024 - 18th May Morning Shift
30
$\int_{\log 4}^{\log 4} \frac{e^{2 x}+e^x}{e^{2 r}-5 e^x+6} d x=$
AP EAPCET 2024 - 18th May Morning Shift
31
$\int_1^2 \frac{x^4-1}{x^6-1} d x=$
AP EAPCET 2024 - 18th May Morning Shift
32

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 $$\int_\limits0^{5 T} f(2 x) d x=$$

AP EAPCET 2022 - 5th July Morning Shift
33

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

AP EAPCET 2022 - 5th July Morning Shift
34

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

AP EAPCET 2022 - 5th July Morning Shift
35

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

then $$f(x)=$$

AP EAPCET 2022 - 5th July Morning Shift
36

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

AP EAPCET 2022 - 4th July Evening Shift
37

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

AP EAPCET 2022 - 4th July Evening Shift
38

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

AP EAPCET 2022 - 4th July Evening Shift
39

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

AP EAPCET 2022 - 4th July Evening Shift
40

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

AP EAPCET 2022 - 4th July Morning Shift
41

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, $$\int_\alpha^\beta x(g \circ f(x)) d x=$$

AP EAPCET 2022 - 4th July Morning Shift
42

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

AP EAPCET 2022 - 4th July Morning Shift
43

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

AP EAPCET 2021 - 20th August Morning Shift
44

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

AP EAPCET 2021 - 20th August Morning Shift
45

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

AP EAPCET 2021 - 19th August Evening Shift
46

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

AP EAPCET 2021 - 19th August Evening Shift
47

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

AP EAPCET 2021 - 19th August Morning Shift
48

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

AP EAPCET 2021 - 19th August Morning Shift
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