1
WB JEE 2024
MCQ (Single Correct Answer)
+1
-0.25

$$\text { If } \int \frac{\log _e\left(x+\sqrt{1+x^2}\right)}{\sqrt{1+x^2}} \mathrm{~d} x=\mathrm{f}(\mathrm{g}(x))+\mathrm{c} \text { then }$$

A
$$\mathrm{f}(x)=\frac{x^2}{2}, \mathrm{~g}(x)=\log _{\mathrm{e}}\left(x+\sqrt{1+x^2}\right)$$
B
$$\mathrm{f}(x)=\log _{\mathrm{e}}\left(x+\sqrt{1+x^2}\right), \mathrm{g}(x)=\frac{x^2}{2}$$
C
$$\mathrm{f}(x)=x^2, \mathrm{~g}(x)=\log _{\mathrm{e}}\left(x+\sqrt{1+x^2}\right)$$
D
$$\mathrm{f}(x)=\log _{\mathrm{e}}\left(x-\sqrt{1+x^2}\right), \mathrm{g}(x)=x^2$$
2
WB JEE 2023
MCQ (Single Correct Answer)
+1
-0.25

If $$I = \int {{{{x^2}dx} \over {{{(x\sin x + \cos x)}^2}}} = f(x) + \tan x + c}$$, then $$f(x)$$ is

A
$${{\sin x} \over {x\sin x + \cos x}}$$
B
$${1 \over {{{(x\sin x + \cos x)}^2}}}$$
C
$${{ - x} \over {\cos x(x\sin x + \cos x)}}$$
D
$${1 \over {\sin x(x\cos x + \sin x)}}$$
3
WB JEE 2023
MCQ (Single Correct Answer)
+1
-0.25

If $$\int {{{dx} \over {(x + 1)(x - 2)(x - 3)}} = {1 \over k}{{\log }_e}\left\{ {{{|x - 3{|^3}|x + 1|} \over {{{(x - 2)}^4}}}} \right\} + c}$$, then the value of k is

A
4
B
6
C
8
D
12
4
WB JEE 2022
MCQ (Single Correct Answer)
+1
-0.25

$$I = \int {\cos (\ln x)dx}$$. Then I =

A
$${x \over 2}\{ \cos (\ln x) + \sin (\ln x)\} + c$$ (c denotes constant of integration)
B
$${x^2}\{ \cos (\ln x) - \sin (\ln x)\} + c$$ (c denotes constant of integration)
C
$${x^2}\sin (\ln x) + c$$ (c denotes constant of integration)
D
$$x\cos (\ln x) + c$$ (c denotes constant of integration)
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