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

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

A
$$\frac{5}{2 \sqrt{3}}$$
B
$$\frac{-1}{2 \sqrt{3}}$$
C
$$\frac{-2}{\sqrt{3}}$$
D
$$\frac{\sqrt{3}}{2}$$
2
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.
3
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 $$\log |\sin (x-\alpha)|$$.
4
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.
MHT CET Subjects
Physics
Mechanics
Optics
Electromagnetism
Modern Physics
Chemistry
Physical Chemistry
Inorganic Chemistry
Organic Chemistry
Mathematics
Algebra
Trigonometry
Calculus
Coordinate Geometry
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