Indefinite Integration · Mathematics · AP EAPCET

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MCQ (Single Correct Answer)

1

$\frac{4 x^2+5}{(x-2)^4}=\frac{A}{(x-2)}+\frac{B}{(x-2)^2}+\frac{C}{(x-2)^3}+\frac{D}{(x-2)^4}$, then $\sqrt{\frac{A}{C}+\frac{B}{C}+\frac{D}{C}}$ is equal to

AP EAPCET 2024 - 23th May Morning Shift
2
If $\int \frac{\sqrt[4]{x}}{\sqrt{x}+\sqrt[4]{x}} d x=$ $\frac{2}{3}\left[A \sqrt[4]{x^3}+B \sqrt[4]{x^2}+C \sqrt[4]{x}+D \log (1+\sqrt[4]{x})\right]+K$, then $\frac{2}{3}(A+B+C+D)$ is equal to
AP EAPCET 2024 - 23th May Morning Shift
3
$\int(\log x)^m x^n d x$ is equal to
AP EAPCET 2024 - 23th May Morning Shift
4
$\int \sin ^{-1}\left(\sqrt{\frac{x-a}{x}}\right) d x$ is equal to
AP EAPCET 2024 - 23th May Morning Shift
5
If $\int \frac{\sin x \cos x}{\sqrt{\cos ^4 x-\sin ^4 x}} d x=-\frac{f(x)}{2}+c$, then domain of $f(x)$ is
AP EAPCET 2024 - 23th May Morning Shift
6

$$ \text { If } \frac{13 x+43}{2 x^2+17 x+30}=\frac{A}{2 x+5}+\frac{B}{x+6} \text {, then } A+B \text { is equal to } $$

AP EAPCET 2024 - 22th May Evening Shift
7
$\int e^{4 x^2+8 x-4}(x+1) \cos \left(3 x^2+6 x-4\right) d x$ is equal to
AP EAPCET 2024 - 22th May Evening Shift
8
$\int\left[(\log 2 x)^2+2 \log 2 x\right] d x$ is equal to
AP EAPCET 2024 - 22th May Evening Shift
9

If $\int \log \left(6 \sin ^2 x+17 \sin x+12\right) \cos x d x=f(x)+c$, then $f\left(\frac{\pi}{2}\right)$ is equal to

AP EAPCET 2024 - 22th May Evening Shift
10
$\int \frac{1}{\left(1+x^2\right) \sqrt{x^2+2}} d x$ is equal to
AP EAPCET 2024 - 22th May Evening Shift
11
$\int \sin ^4 x \cos ^4 x d x$ is equal to
AP EAPCET 2024 - 22th May Evening Shift
12
$$ \int \frac{x^2-1}{x^3 \sqrt{2 x^4-2 x^2+1}} d x $$
AP EAPCET 2024 - 22th May Morning Shift
13

$$ \int \frac{x^3 \tan ^{-1} x^4}{1+x^8} d x= $$

AP EAPCET 2024 - 22th May Morning Shift
14
$$ \int \frac{2}{1+x+x^2} d x= $$
AP EAPCET 2024 - 22th May Morning Shift
15

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

AP EAPCET 2024 - 22th May Morning Shift
16

$$ \int \frac{\sin 7 x}{\sin 2 x \sin 5 x} d x= $$

AP EAPCET 2024 - 22th May Morning Shift
17
If $\frac{x+2}{\left(x^2+3\right)\left(x^4+x^2\right)\left(x^2+2\right)}=\frac{A x+B}{x^2+3}+\frac{C x+D}{x^2+2}$ $+\frac{E x^3+F x^2+G x+H}{x^4+x^2}$, then $(E+F)(C+D)(A)=$
AP EAPCET 2024 - 21th May Evening Shift
18
$\int \frac{\sin ^6 x}{\cos ^8 x} d x=$
AP EAPCET 2024 - 21th May Evening Shift
19
$\int \frac{x^5}{x^2+1} d x=$
AP EAPCET 2024 - 21th May Evening Shift
20
$$\int {\left( {\sum\limits_{r = 0}^\infty {{{{x^r}{3^r}} \over {r!}}} } \right)dx = } $$
AP EAPCET 2024 - 21th May Evening Shift
21
$\int \frac{x^4+1}{x^6+1} d x=$
AP EAPCET 2024 - 21th May Evening Shift
22
$\int e^x(x+1)^2 d x=$
AP EAPCET 2024 - 21th May Evening Shift
23

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

AP EAPCET 2024 - 21th May Morning Shift
24
If $x \in\left[2 n \pi-\frac{\pi}{4}, 2 n \pi+\frac{3 \pi}{4}\right]$ and $n \in Z$, then $\int \sqrt{1-\sin 2 x} d x=$
AP EAPCET 2024 - 21th May Morning Shift
25
$\int e^x\left(\frac{x+2}{x+4}\right)^2 d x=$
AP EAPCET 2024 - 21th May Morning Shift
26
If $\int \frac{1}{1-\cos x} d x=\tan \left(\frac{x}{\alpha}+\beta\right)+c$, then one of the values of $\frac{\pi \alpha}{4}-\beta$ is
AP EAPCET 2024 - 21th May Morning Shift
27
If $n \geq 2$ is a natural number and $0<\theta<\frac{\pi}{2}$, then $\int \frac{\left(\cos ^n \theta-\cos \theta\right)^{1 / n}}{\cos ^{n+1} \theta} \sin \theta d \theta=$
AP EAPCET 2024 - 21th May Morning Shift
28
If $\frac{x^2+3}{x^4+2 x^2+9}=\frac{A x+B}{x^2+a x+b}+\frac{C x+D}{x^2+c x+b}$, then $a A+b B+c C+D=$
AP EAPCET 2024 - 20th May Evening Shift
29
$\int \frac{d x}{x\left(x^4+1\right)}=$
AP EAPCET 2024 - 20th May Evening Shift
30
$\int \frac{d x}{\sqrt{\sin ^3 x \cos (x-a)}}=$
AP EAPCET 2024 - 20th May Evening Shift
31
$\int \frac{e^{2 x}}{\sqrt[4]{e^x+1}} d x=$
AP EAPCET 2024 - 20th May Evening Shift
32
$\int \frac{2-\sin x}{2 \cos x+3} d x=$
AP EAPCET 2024 - 20th May Evening Shift
33
$\int \sin ^{-1} \sqrt{\frac{x}{a+x}} d x=$
AP EAPCET 2024 - 20th May Evening Shift
34
If $\frac{A}{x-a}+\frac{B x+C}{x^2+b^2}=\frac{1}{(x-a)\left(x^2+b^2\right)}$, then $\mathrm{C}=$
AP EAPCET 2024 - 20th May Morning Shift
35
$\int \frac{2 x^2-3}{\left(x^2-4\right)\left(x^2+1\right)} d x=A \tan ^{-1} x+B \log (x-2)+C \log (x+2)$, then $6 A+7 B-5 C=$
AP EAPCET 2024 - 20th May Morning Shift
36
$\int \frac{3 x^9+7 x^8}{\left(x^2+2 x+5 x^8\right)^2} d x=$
AP EAPCET 2024 - 20th May Morning Shift
37
$\int \frac{\cos x+x \sin x}{x(x+\cos x)} d x=$
AP EAPCET 2024 - 20th May Morning Shift
38
If $\int \sqrt{\frac{2}{1+\sin x}} d x=2 \log |A(x)-B(x)|+C$ and $0 \leq x \leq \frac{\pi}{2}$, then $B\left(\frac{\pi}{4}\right)=$
AP EAPCET 2024 - 20th May Morning Shift
39

$$ \begin{aligned} &\text { If } \int \frac{3}{2 \cos ^3 x \sqrt{2 \sin 2 x}} d x=\frac{3}{2}(\tan x)^B+\frac{3}{10}(\tan x)^A+C \text {, than }\\&A= \end{aligned} $$

AP EAPCET 2024 - 20th May Morning Shift
40
If $\frac{1}{x^4+1}=\frac{A x+B}{x^2+\sqrt{2} x+1}+\frac{C x+D}{x^2-\sqrt{2} x+1}$, then $B D-A C=$
AP EAPCET 2024 - 19th May Evening Shift
41
$$ \int \frac{2 x^2 \cos x^2-\sin x^2}{x^2} d x= $$
AP EAPCET 2024 - 19th May Evening Shift
42
If $\int \frac{\log \left(1+x^4\right)}{x^3} d x=f(x) \log \left(\frac{1}{g(x)}\right)+\tan ^{-1}$ $(h(x))+c$, then $h(x)\left[f(x)+f\left(\frac{1}{x}\right)\right]=$
AP EAPCET 2024 - 19th May Evening Shift
43
Let $f(x)=\int \frac{x}{\left(x^2+1\right)\left(x^2+3\right)} d x$. If $f(3)=\frac{1}{4} \log \left(\frac{5}{6}\right)$, then $f(0)=$
AP EAPCET 2024 - 19th May Evening Shift
44
$$ \int \frac{2 \cos 2 x}{(1+\sin 2 x)(1+\cos 2 x)} d x= $$
AP EAPCET 2024 - 19th May Evening Shift
45
$$ \int\left(\frac{x}{x \cos x-\sin x}\right)^2 d x= $$
AP EAPCET 2024 - 19th May Evening Shift
46
$\int \frac{1}{x^5 \sqrt[3]{x^3+1}} d x=$
AP EAPCET 2024 - 18th May Morning Shift
47
$\int \frac{x+1}{\sqrt{x^2+x+1}} d x=$
AP EAPCET 2024 - 18th May Morning Shift
48
$\int\left(\tan ^9 x+\tan x\right) d x=0$
AP EAPCET 2024 - 18th May Morning Shift
49
$\int \frac{\operatorname{cosec} x}{3 \cos x+4 \sin x} d x=$
AP EAPCET 2024 - 18th May Morning Shift
50
$\int e^{2 x+3} \sin 6 x d x=$
AP EAPCET 2024 - 18th May Morning Shift
51

$$\frac{2 x^2+1}{x^3-1}=\frac{A}{x-1}+\frac{B x+C}{x^2+x+1} \Rightarrow 7 A+2 B+C=$$

AP EAPCET 2022 - 5th July Morning Shift
52

$$\int \frac{3 x+4}{x^3-2 x+4} d x=\log f(x)+C \Rightarrow f(3)=$$

AP EAPCET 2022 - 5th July Morning Shift
53

$$\int \frac{e^{\tan ^{-1} x}}{1+x^2}\left[\left(\sec ^{-1} \sqrt{1+x^2}\right)^2+\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right] d x=$$

AP EAPCET 2022 - 5th July Morning Shift
54

$$\int \frac{d x}{(x-3)^{\frac{4}{5}}(x+1)^{\frac{6}{5}}}=$$

AP EAPCET 2022 - 5th July Morning Shift
55

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

AP EAPCET 2022 - 5th July Morning Shift
56

If $$f(x)=\int x^2 \cos ^2 x\left(2 x \tan ^2 x-2 x-6 \tan x\right) d x$$ and $$f(0)=\pi$$, then $$f(x)=$$

AP EAPCET 2022 - 4th July Evening Shift
57

If $$\int \frac{e^{\sqrt{x}}}{\sqrt{x}}(x+\sqrt{x}) d x=e^{\sqrt{x}}[A x+B \sqrt{x}+C]+K$$ then $$A+B+C=$$

AP EAPCET 2022 - 4th July Evening Shift
58

If $$\int \frac{1+\sqrt{\tan x}}{\sin 2 x} d x=A \log \tan x+B \tan x+C$$, then $$4 A-2 B=$$

AP EAPCET 2022 - 4th July Evening Shift
59

$$\int \frac{1+\tan x \tan (x+a)}{\tan x \tan (x+a)} d x=$$

AP EAPCET 2022 - 4th July Evening Shift
60

Assertion (A) If $$I_n=\int \cot ^n x d x$$, then $$I_6+I_4=\frac{-\cot ^5 x}{5}$$

Reason (R) $$\int \cot ^n x d x=\frac{-\cot ^{n-1} x}{n} -\int \cot ^{n-2} x d x$$

AP EAPCET 2022 - 4th July Morning Shift
61

If $$I_n=\int \tan ^n x d x$$, and $$I_0+I_1+2 I_2+2 I_3+2 I_4 +I_5+I_6=\sum_\limits{k=1}^n \frac{\tan ^k x}{k}$$, then $$n=$$

AP EAPCET 2022 - 4th July Morning Shift
62

$$\int \frac{e^{\cot x}}{\sin ^2 x}(2 \log \operatorname{cosec} x+\sin 2 x) d x=$$

AP EAPCET 2022 - 4th July Morning Shift
63

The parametric form of a curve is $$x=\frac{t^3}{t^2-1} y=\frac{t}{t^2-1}$$, then $$\int \frac{d x}{x-3 y}=$$

AP EAPCET 2022 - 4th July Morning Shift
64

Given, $$\frac{3 x-2}{(x+1)^2(x+3)}=\frac{A}{x+1} +\frac{B}{(x+1)^2}+\frac{C}{x+3}$$, then $$4 A+2 B+4 C$$ is equal to

AP EAPCET 2021 - 20th August Morning Shift
65

$$\int \frac{\sin \alpha}{\sqrt{1+\cos \alpha}} d \alpha$$ is equal to

AP EAPCET 2021 - 20th August Morning Shift
66

If $$\int \frac{\cos 4 x+1}{\cot x-\tan x}=k \cos 4 x+C$$, then $$k$$ is equal to

AP EAPCET 2021 - 20th August Morning Shift
67

If $$\int\left[\cos (x) \cdot \frac{d}{d x}(\operatorname{cosec}(x)] d x=f(x)+g(x)+c\right.$$ then $$f(x) \cdot g(x)$$ is equal to

AP EAPCET 2021 - 20th August Morning Shift
68

If $$\int \frac{(2 x+1)^6}{(3 x+2)^8} d x=P\left(\frac{2 x+1}{3 x+2}\right)^Q+R$$, then $$\frac{P}{Q}$$ is equal to

AP EAPCET 2021 - 20th August Morning Shift
69

Which of the following is partial fraction of $$\frac{-x^2+6 x+13}{(3 x+5)\left(x^2+4 x+4\right)}$$ is equal to

AP EAPCET 2021 - 19th August Evening Shift
70

$$\int \frac{1+x+\sqrt{x+x^2}}{\sqrt{x}+\sqrt{1+x}} d x$$ is equal to

AP EAPCET 2021 - 19th August Evening Shift
71

$$\int(\cos x) \log \cot \left(\frac{x}{2}\right) d x$$ is equal to

AP EAPCET 2021 - 19th August Evening Shift
72

$$\int \sqrt{e^{4 x}+e^{2 x}} d x$$ is equal to

AP EAPCET 2021 - 19th August Evening Shift
73

If $$\int \frac{1}{1+\sin x} d x=\tan (f(x))+c$$, then $$f^{\prime}(0)$$ is equal to

AP EAPCET 2021 - 19th August Evening Shift
74

$$\int \frac{e^x(x+3)}{(x+5)^3} d x$$ is equal to

AP EAPCET 2021 - 19th August Morning Shift
75

If $$\int \frac{(x-1)^2}{\left(x^2+1\right)^2} d x=\tan ^{-1}(x)+g(x)+k$$, then $$g(x)$$ is equal to

AP EAPCET 2021 - 19th August Morning Shift
76

If $$\int \frac{1-(\cot x)^{2021}}{\tan x+(\cot x)^{2022}} d x=\frac{1}{A} \log\left|(\sin x)^{2023}+(\cos x)^{2023}\right|+c$$, then $$A$$ is equal to

AP EAPCET 2021 - 19th August Morning Shift
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