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

$$\int \frac{5 \tan x}{\tan x-2} \mathrm{~d} x=x+\mathrm{a} \log |\sin x-2 \cos x|+\mathrm{c},$$ (where $$c$$ is a constant of integration), then the value of $$a$$ is

A
1
B
$$\frac{1}{2}$$
C
2
D
3
2
MHT CET 2023 10th May Evening Shift
+2
-0

The value of $$\int \frac{\left(x^2-1\right) d x}{x^3 \sqrt{2 x^4-2 x^2+1}}$$ is

A
$$2 \sqrt{2-\frac{2}{x^2}+\frac{1}{x^4}}+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
B
$$2 \sqrt{2+\frac{2}{x^2}+\frac{1}{x^4}}+c$$, where $$\mathrm{c}$$ is a constant of integration.
C
$$\frac{1}{2} \sqrt{2-\frac{2}{x^2}+\frac{1}{x^4}}+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
D
$$2 \sqrt{2-\frac{2}{x^2}-\frac{1}{x^4}}+c$$, where $$\mathrm{c}$$ is a constant of integration.
3
MHT CET 2023 10th May Evening Shift
+2
-0

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

A
$$\mathrm{e}^x \cdot \cot x+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
B
$$\mathrm{e}^x \cdot \operatorname{cosec} x+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
C
$$-\mathrm{e}^x \cdot \cot x+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
D
$$-\mathrm{e}^x \cdot \operatorname{cosec} x+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
4
MHT CET 2023 10th May Morning Shift
+2
-0

If $$\int \sqrt{\frac{x-7}{x-9}} d x=A \sqrt{x^2-16 x+63}+\log \left|(x-8)+\sqrt{x^2-16 x+63}\right|+c,$$

(where $$\mathrm{c}$$ is a constant of integration) then $$\mathrm{A}$$ is

A
$$-1$$
B
$$\frac{1}{2}$$
C
$$1$$
D
$$\frac{-1}{2}$$
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