1
MHT CET 2023 14th May Morning Shift
+2
-0

$$\int \frac{\sin 2 x\left(1-\frac{3}{2} \cos x\right)}{e^{\sin ^2 x+\cos ^3 x}} d x=$$

A
$$\mathrm{e}^{\sin ^2 x+\cos ^3 x}+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
B
$$\mathrm{-e}^{-(\sin ^2 x+\cos ^3 x)}+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
C
$$\mathrm{e}^{-(\sin ^2 x+\cos ^3 x)^2}+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
D
$$\mathrm{e}^{\sin ^2 x+\cos x}+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
2
MHT CET 2023 14th May Morning Shift
+2
-0

If $$\int \frac{\cos \theta}{5+7 \sin \theta-2 \cos ^2 \theta} d \theta=A \log _e|f(\theta)|+c$$ (where $$c$$ is a constant of integration), then $$\frac{f(\theta)}{A}$$ can be

A
$$\frac{2 \sin \theta+1}{\sin \theta+3}$$
B
$$\frac{2 \sin \theta+1}{5(\sin \theta+3)}$$
C
$$\frac{5(\sin \theta+3)}{2 \sin \theta+1}$$
D
$$\frac{5(2 \sin \theta+1)}{\sin \theta+3}$$
3
MHT CET 2023 14th May Morning Shift
+2
-0

$$\int \frac{\sin x+\sin ^3 x}{\cos 2 x} d x=A \cos x+B \log \mathrm{f}(x)+c$$ (where $$\mathrm{c}$$ is a constant of integration). Then values of $$\mathrm{A}, \mathrm{B}$$ and $$\mathrm{f}(x)$$ are

A
$$\mathrm{A}=\frac{1}{2}, \mathrm{~B}=\frac{-3}{4 \sqrt{2}}, \mathrm{f}(x)=\frac{\sqrt{2} \cos x-1}{\sqrt{2} \cos x+1}$$
B
$$A=-\frac{1}{2}, B=\frac{-3}{4 \sqrt{2}}, \mathrm{f}(x)=\frac{\sqrt{2} \cos x+1}{\sqrt{2} \cos x-1}$$
C
$$\mathrm{A}=\frac{1}{2}, \mathrm{~B}=\frac{-3}{4 \sqrt{2}}, \mathrm{f}(x)=\frac{\sqrt{2} \cos x+1}{\sqrt{2} \cos x-1}$$
D
$$\mathrm{A}=\frac{3}{2}, \mathrm{~B}=\frac{1}{2}, \mathrm{f}(x)=\frac{\sqrt{2} \cos x-1}{\sqrt{2} \cos x+1}$$
4
MHT CET 2023 14th May Morning Shift
+2
-0

If $$\int \frac{x^3 \mathrm{~d} x}{\sqrt{1+x^2}}=\mathrm{a}\left(1+x^2\right) \sqrt{1+x^2}+\mathrm{b} \sqrt{1+x^2}+\mathrm{c}$$ (where $$\mathrm{c}$$ is a constant of integration), then the value of $$3 \mathrm{ab}$$ is

A
$$-$$3
B
$$-$$1
C
1
D
3
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