Indefinite Integration PYQs - Last 10 Years

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Practice 267 MHT CET Mathematics questions from Indefinite Integration. Use the year-wise and type-wise breakdown to prioritize recent PYQs, then continue into the question list below.

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Last 10 Years Indefinite Integration Questions

Showing 50 of 267 filtered questions.

1Indefinite Integration

If $\mathrm{f}(x)=\frac{x}{x+1}, x \neq-1$ and (fof) $(x)=\mathrm{F}(x)$, then $\int \mathrm{F}(x) \mathrm{d} x$ is

2Indefinite Integration

$\int\left(1+x-\frac{1}{x}\right) \mathrm{e}^{x+\frac{1}{x}} \mathrm{~d} x$ is equal to

3Indefinite Integration

If $\int \mathrm{e}^{x^2} \cdot x^3 \mathrm{dx}=\mathrm{e}^{x^2} \mathrm{f}(x)+\mathrm{c}$ and $\mathrm{f}(1)=0$ (where c is a constant of integration), then the value of $f(x)$ is

4Indefinite Integration

$$\begin{aligned} & \text { If } \\ & \int(7 x-2) \sqrt{3 x+2} \mathrm{~d} x=\mathrm{A}(3 x+2)^{\frac{5}{2}}+\mathrm{B}(3 x+2)^{\frac{3}{2}}+\mathrm{c} \end{aligned}$$
(where c is a constant of integration), then the values of $A$ and $B$ a...

5Indefinite Integration

If $x \in[-1,1]$, then the value of $\int \mathrm{e}^{\sin ^{-1} x}\left(\frac{x+\sqrt{1-x^2}}{\sqrt{1-x^2}}\right) \mathrm{d} x$ is

6Indefinite Integration

The value of $\int \frac{\cos ^3 x}{\sin ^2 x+\sin x} \mathrm{~d} x$ is

7Indefinite Integration

\(\int \frac{x \mathrm{~d} x}{(x-1)^2(x+2)}=\)

8Indefinite Integration

\(\int \frac{2 x+5}{\sqrt{7-6 x-x^2}} d x=A \sqrt{7-6 x-x^2}+B \sin ^{-1}\left(\frac{x+3}{4}\right)+\mathrm{c}\)

(where c is a constant of integration) then the value of $A+B$ is

9Indefinite Integration

\(\int \cos (\log x) \mathrm{d} x=\)

10Indefinite Integration

If $\mathrm{f}(x)=1+x ; \mathrm{g}(x)=\log x$, then $\int \mathrm{g}(\mathrm{f}(x)) \mathrm{d} x$ is equal to

11Indefinite Integration

\(\int \frac{x+1}{x\left(1+x \mathrm{e}^x\right)^2} \mathrm{dx}=\)

12Indefinite Integration

\(\int \frac{\sqrt{x}}{x+1} d x=\)

13Indefinite Integration

\(\int \frac{1+\sin (\log x)}{1+\cos (\log x)} d x=\)

14Indefinite Integration

If $\quad \int(2 x+4) \sqrt{x-1} \mathrm{~d} x=\mathrm{a}(x-1)^{\frac{5}{2}}+\mathrm{b}(x-1)^{\frac{3}{2}}+\mathrm{c}$, (where c is a constant of integration), then the value of $a+b$ is

15Indefinite Integration

If, $\int \frac{d \theta}{\cos ^2 \theta(\tan 2 \theta+\sec 2 \theta)}=\lambda \tan \theta+2 \log _{\mathrm{e}}|\mathrm{f}(\theta)|+\mathrm{c}$ (where c is a constant of integration), then the ordered pair $(\lambda,|f(\theta)|)$ is equal t...

16Indefinite Integration

\(\int \sin ^{-1}\left(\frac{2 x}{1+x^2}\right) \mathrm{d} x=\)

17Indefinite Integration

The value of $\int \frac{\mathrm{d} x}{x^2\left(x^4+1\right)^{\frac{3}{4}}}$ is

18Indefinite Integration

$\int \frac{\mathrm{d} x}{\sqrt{\mathrm{e}^x-1}}=2 \tan ^{-1}(\mathrm{f}(x))+\mathrm{c}$ where $x>0$ and c is a constant of integration, then $\mathrm{f}(x)$ is

19Indefinite Integration

\(\int \frac{1}{\sin (x-a) \sin x} d x=\)

20Indefinite Integration

If \(I=\int \frac{\sin x+\sin ^3 x}{\cos 2 x} d x=P \cos x+Q \log \left|\frac{\sqrt{2} \cos x-1}{\sqrt{2} \cos x+1}\right|\) (where \(c\) is a constant of integration), then values of \(\mathrm{P}\) and \(\mathrm{Q}\) are respectively

21Indefinite Integration

\(\int \frac{\mathrm{e}^{\tan ^{-1} x}}{1+x^2}\left[\left(\sec ^{-1} \sqrt{1+x^2}\right)^2+\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right] \mathrm{d} x, x > 0=\)

22Indefinite Integration

\(\int \frac{x+1}{x\left(1+x \mathrm{e}^x\right)^2} \mathrm{~d} x=\)

23Indefinite Integration

Let \(\alpha \in\left(0, \frac{\pi}{2}\right)\) be fixed. If the integral
\(\int \frac{\tan x+\tan \alpha}{\tan x-\tan \alpha} \mathrm{d} x=\mathrm{A}(x) \cos 2 \alpha+\mathrm{B}(x) \sin 2 \alpha+\mathrm{c},\)
(where \(\mathrm{c}\) is a con...

24Indefinite Integration

\(\int(\sqrt{\tan x}+\sqrt{\cot x}) d x=\)

25Indefinite Integration

If \(\int \frac{\sin x}{3+4 \cos ^2 x} \mathrm{~d} x=\mathrm{A} \tan ^{-1}(\mathrm{~B} \cos x)+\mathrm{c}\), (where \(\mathrm{c}\) is a constant of integration), then the value of \(\mathrm{A}+\mathrm{B}\) is

26Indefinite Integration

Let \(f(x)=\int \frac{x^2-3 x+2}{x^4+1} \mathrm{~d} x\), then function decreases in the interval

27Indefinite Integration

If \(\int \frac{x^3 \mathrm{~d} x}{\sqrt{1+x^2}}=\mathrm{a}\left(1+x^2\right) \sqrt{1+x^2}+\mathrm{b} \sqrt{1+x^2}+\mathrm{c}\)
(where \(\mathrm{c}\) is a constant of integration), then the value of \(3 \mathrm{ab}\) is

28Indefinite Integration

\(\int \frac{\sin x+\sin ^3 x}{\cos 2 x} d x=A \cos x+B \log \mathrm{f}(x)+c\)
(where \(\mathrm{c}\) is a constant of integration). Then values of \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{f}(x)\) are

29Indefinite Integration

If \(\int \frac{\cos \theta}{5+7 \sin \theta-2 \cos ^2 \theta} d \theta=A \log _e|f(\theta)|+c\) (where \(c\) is a constant of integration), then \(\frac{f(\theta)}{A}\) can be

30Indefinite Integration

\(\int \frac{\sin 2 x\left(1-\frac{3}{2} \cos x\right)}{e^{\sin ^2 x+\cos ^3 x}} d x=\)

31Indefinite Integration

If \(I=\int \frac{d x}{\sin (x-a) \sin (x-b)}\), then I is given by

32Indefinite Integration

\(\int \frac{\mathrm{d} x}{\cot ^2 x-1}=\frac{1}{\mathrm{~A}} \log |\sec 2 x+\tan 2 x|-\frac{x}{\mathrm{~B}}+\mathrm{c}\), (where \(\mathrm{c}\) is constant of integration), then \(\mathrm{A}+\mathrm{B}=\)

33Indefinite Integration

If \(\int \frac{x^2}{\sqrt{1-x}} \mathrm{~d} x=\mathrm{p} \sqrt{1-x}\left(3 x^2+4 x+8\right)+\mathrm{c}\) where \(\mathrm{c}\) is a constant of integration, then the value of \(p\) is

34Indefinite Integration

The value of \(\int \mathrm{e}^x\left(\frac{x^2+4 x+4}{(x+4)^2}\right) \mathrm{d} x\) is :

35Indefinite Integration

If \(\mathrm{f}(x)=\int \frac{x^2 \mathrm{~d} x}{\left(1+x^2\right)\left(1+\sqrt{1+x^2}\right)}\) and \(\mathrm{f}(0)=0\), then \(\mathrm{f}(1)\) is

36Indefinite Integration

If \(\int x^5 e^{-4 x^3} \mathrm{~d} x=\frac{1}{48} \mathrm{e}^{-4 x^3} \mathrm{f}(x)+\mathrm{c}\), where \(\mathrm{c}\) is a constant of integration, then \(\mathrm{f}(x)\) is given by

37Indefinite Integration

\(\int \frac{\log \left(x^2+a^2\right)}{x^2} d x=\)

38Indefinite Integration

If \(\mathrm{I}=\int \frac{2 x-7}{\sqrt{3 x-2}} \mathrm{~d} x\), then \(\mathrm{I}\) is given by

39Indefinite Integration

If \(I=\int \frac{e^x}{e^{4 x}+e^{2 x}+1} d x\) and
\(J=\int \frac{e^{-x}}{e^{-4 x}+e^{-2 x}+1} d x\), then for any arbitrary constant \(C\), than the value of \(J-I\) equals

40Indefinite Integration

\(\int \frac{2+\cos \frac{x}{2}}{x+\sin \frac{x}{2}} d x=\)

41Indefinite Integration

\(\int \frac{x-3}{(x-1)^3} e^x d x=\)

42Indefinite Integration

If \(\int \frac{\log \left(t+\sqrt{1+t^2}\right)}{\sqrt{1+t^2}} d t=\frac{1}{2}[g(t)]^2+c\), (where \(c\) is a constant of integration), then \(g(2)\) is

43Indefinite Integration

If \(\int \cos ^{\frac{3}{5}} x \cdot \sin ^3 x d x=\frac{-1}{m} \cos ^m x+\frac{1}{n} \cos ^n x+c\), (where \(\mathrm{c}\) is the constant of integration), then \((\mathrm{m}, \mathrm{n})=\)

44Indefinite Integration

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

45Indefinite Integration

The integral \(\int \frac{\sin ^2 x \cos ^2 x}{\left(\sin ^5 x+\cos ^3 x \sin ^2 x+\sin ^3 x \cos ^2 x+\cos ^5 x\right)^2} \mathrm{~d} x\) is equal to

46Indefinite Integration

\(\int \frac{\operatorname{cosec} x d x}{\cos ^2\left(1+\log \tan \frac{x}{2}\right)}=\)

47Indefinite Integration

\(\int \frac{1}{(x+2)(1+x)^2} d x\) has the value

48Indefinite Integration

\(\int\left(\frac{\tan \left(\frac{1}{x}\right)}{x}\right)^2 d x=\)

49Indefinite Integration

If \(\int \frac{\sqrt{1-x^2}}{x^4} \mathrm{~d} x=\mathrm{A}(x)\left(\sqrt{1-x^2}\right)^{\mathrm{m}}+\mathrm{c}\) for a suitable chosen integer \(\mathrm{m}\) and a function \(\mathrm{A}(x)\), where \(\mathrm{c}\) is a constant of integrati...

50Indefinite Integration

\(\int \frac{1}{\cos ^3 x \sqrt{\sin 2 x}} d x=\)