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

If $$\int_\limits0^{\frac{1}{2}} \frac{x^2}{\left(1-x^2\right)^{\frac{3}{2}}} \mathrm{~d} x=\frac{\mathrm{k}}{6}$$, then the value of $$\mathrm{k}$$ is

A
$$2 \sqrt{3}-\pi$$
B
$$2 \sqrt{3}+\pi$$
C
$$3 \sqrt{2}+\pi$$
D
$$3 \sqrt{2}-\pi$$
2
MHT CET 2023 9th May Evening Shift
+2
-0

$$\int_\limits1^2 \frac{\mathrm{d} x}{\left(x^2-2 x+4\right)^{\frac{3}{2}}}=\frac{\mathrm{k}}{\mathrm{k}+5} \text {, then } \mathrm{k} \text { has the value }$$

A
1
B
2
C
$$-$$1
D
$$-$$2
3
MHT CET 2023 9th May Morning Shift
+2
-0

If $$\mathrm{f}(x)$$ is a function satisfying $$\mathrm{f}^{\prime}(x)=\mathrm{f}(x)$$ with $$\mathrm{f}(0)=1$$ and $$\mathrm{g}(x)$$ is a function that satisfies $$\mathrm{f}(x)+\mathrm{g}(x)=x^2$$. Then the value of the integral $$\int_\limits0^1 f(x) g(x) d x$$ is

A
$$e-\frac{e^2}{2}-\frac{5}{2}$$
B
$$\mathrm{e}+\frac{\mathrm{e}^2}{2}-\frac{3}{2}$$
C
$$\mathrm{e}-\frac{\mathrm{e}^2}{2}-\frac{3}{2}$$
D
$$\mathrm{e}+\frac{\mathrm{e}^2}{2}+\frac{5}{2}$$
4
MHT CET 2022 11th August Evening Shift
+2
-0

$$\int_\limits{\frac{\pi}{4}}^{\frac{3 \pi}{4}} \frac{\mathrm{d} x}{1+\cos x}$$ is equal to

A
$$-2 \sqrt{2}$$
B
$$2$$
C
$$-2-2 \sqrt{2}$$
D
$$-2$$
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