1
MHT CET 2023 10th May Morning Shift
+2
-0

$$\int \frac{d x}{\sin x+\cos x}=$$

A
$$\sqrt{2} \log \tan \left(x+\frac{\pi}{4}\right)+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
B
$$\frac{1}{\sqrt{2}} \log \tan \left(\frac{x}{2}+\frac{\pi}{8}\right)+c$$, where c is a constant of integration.
C
$$\frac{1}{\sqrt{2}} \log \left(\frac{\tan \frac{x}{2}-\sqrt{2}+1}{\tan \frac{x}{2}+\sqrt{2}+1}\right)+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
D
$$-\frac{1}{\sqrt{2}} \log \left(\frac{\tan \frac{x}{2}-(\sqrt{2}+1)}{\tan \frac{x}{2}+\sqrt{2}-1}\right)+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
2
MHT CET 2023 10th May Morning Shift
+2
-0

If $$\mathrm{I}=\int \frac{\mathrm{d} x}{x^2\left(x^4+1\right)^{\frac{3}{4}}}$$, then $$\mathrm{I}$$ is

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