1
MHT CET 2023 9th May Morning Shift
+2
-0

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

A
$$\left(\tan ^{-1} x\right)^2 \mathrm{e}^{\tan ^{-1} x}+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
B
$$\left(\tan ^{-1} x\right) \mathrm{e}^{\tan ^{-1} x}+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
C
$$\left(\tan ^{-1} x\right) \mathrm{e}^{2 \tan ^{-1} x}+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
D
$$\left(\tan ^{-1} x\right)^2 \mathrm{e}^{2 \tan ^{-1} x}+c$$, where $$\mathrm{c}$$ is a constant of integration.
2
MHT CET 2023 9th May Morning Shift
+2
-0

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

A
$$\frac{1}{2}, \frac{3}{4 \sqrt{2}}$$
B
$$\frac{1}{2}, \frac{-3}{4 \sqrt{2}}$$
C
$$\frac{1}{2}, \frac{3}{2 \sqrt{2}}$$
D
$$\frac{1}{2}, \frac{-3}{2 \sqrt{2}}$$
3
MHT CET 2023 9th May Morning Shift
+2
-0

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

A
$$\sin \mathrm{a}(\log (\sin (x-\mathrm{a}) \cdot \operatorname{cosec} x))+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
B
$$\operatorname{cosec} a(\log (\sin (x-a) \cdot \operatorname{cosec} x))+c$$, where $$\mathrm{c}$$ is a constant of integration.
C
$$-\sin \mathrm{a}(\log (\sin (x-\mathrm{a}) \cdot \sin x))+\mathrm{c}$$, where c is a constant of integration.
D
$$-\operatorname{cosec} \mathrm{a}(\log (\sin (x-\mathrm{a}) \cdot \sin x))+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
4
MHT CET 2022 11th August Evening Shift
+2
-0

If $$f(x)=\sqrt{\tan x}$$ and $$g(x)=\sin x \cdot \cos x$$ then $$\int \frac{f(x)}{g(x)} \mathrm{d} x$$ is equal to (where $$C$$ is a constant of integration)

A
$$2 \sqrt{\tan x}+C$$
B
$$\frac{1}{2} \sqrt{\tan x}+C$$
C
$$\sqrt{\tan x}+C$$
D
$$\frac{3}{2} \sqrt{\tan x}+C$$
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