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

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 constant of integration), then functions $$\mathrm{A}(x)$$ and $$\mathrm{B}(x)$$ are respectively

A
$$x+\alpha$$ and $$\log |\sin (x+\alpha)|$$.
B
$$x-\alpha$$ and $$\log |\sin (x-\alpha)|$$.
C
$$x-\alpha$$ and $$\log |\cos (x-\alpha)|$$.
D
$$x+\alpha$$ and $$\log |\sin (x-\alpha)|$$.
2
MHT CET 2023 9th May Morning Shift
+2
-0

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

A
$$\log \left|\frac{x \mathrm{e}^x}{1+x \mathrm{e}^x}\right|+c$$, where $$\mathrm{c}$$ is a constant of integration.
B
$$\log \left|\frac{x \mathrm{e}^x}{1+x \mathrm{e}^x}\right|-\frac{1}{1+x \mathrm{e}^x}+\mathrm{c}$$, where c is a constant of integration.
C
$$\log \left|1+x \mathrm{e}^x\right|+\frac{1}{1+x \mathrm{e}^x}+\mathrm{c}$$, where $$\mathrm{c}$$ is constant of integration.
D
$$\log \left|\frac{x \mathrm{e}^x}{1+x \mathrm{e}^x}\right|+\frac{1}{1+x \mathrm{e}^x}+\mathrm{c}$$, where $$\mathrm{c}$$ is constant of integration.
3
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.
4
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
$$\frac{1}{2}, \frac{-3}{2 \sqrt{2}}$$
EXAM MAP
Medical
NEET