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

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

A
$$\sqrt{2} \sin ^{-1}(\sin x-\cos x)+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
B
$$\frac{1}{\sqrt{2}} \sin ^{-1}(\sin x-\cos x)+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
C
$$\sin ^{-1}(\sin x-\cos x)+c$$, where c is a constant of integration.
D
$$2 \sin ^{-1}(\sin x-\cos x)+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
2
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 $$\log |\sin (x-\alpha)|$$.
3
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.
4
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.
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