1
MHT CET 2023 11th May Evening Shift
+2
-0

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

A
$$\log \left(\sec \left(\frac{x^2+1}{2}\right)\right)+$$ c, where c is a constant of integration.
B
$$\log \left(\frac{x^2+1}{2}\right)+c$$, where $$\mathrm{c}$$ is a constant of integration.
C
$$\log \left(\sin \left(\frac{x^2+1}{2}\right)\right)+$$ c, where c is a constant of integration.
D
$$2 \log \left(x^2+1\right)+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
2
MHT CET 2023 11th May Evening Shift
+2
-0

If $$\int \frac{\cos 8 x+1}{\cot 2 x-\tan 2 x} \mathrm{~d} x=\mathrm{A} \cos 8 x+\mathrm{c}$$, where $$\mathrm{c}$$ is an arbitrary constant, then the value of $$\mathrm{A}$$ is

A
$$\frac{1}{16}$$
B
$$\frac{1}{8}$$
C
$$\frac{-1}{8}$$
D
$$\frac{-1}{16}$$
3
MHT CET 2023 11th May Evening Shift
+2
-0

The value of $$\int(1-\cos x) \cdot \operatorname{cosec}^2 x d x$$ is

A
$$\frac{1}{2} \tan \frac{x}{2}+\mathrm{c}$$, where c is a constant of integration.
B
$$\tan \frac{x}{2}+\mathrm{c}$$, where c is a constant of integration.
C
$$2 \cot \frac{x}{2}+\mathrm{c}$$, where c is a constant of integration.
D
$$\cot \frac{x}{2}+\mathrm{c}$$, where c is a constant of integration.
4
MHT CET 2023 11th May Evening Shift
+2
-0

If $$\mathrm{I}=\int \sin (\log (x)) \mathrm{d} x$$, then $$\mathrm{I}$$ is given by

A
$$-\frac{x}{2}(\sin (\log x)-\cos (\log x))+\mathrm{c}$$, where c is a constant of integration.
B
$$\frac{x}{2}(\sin (\log x)-\cos (\log x))+\mathrm{c}$$, where c is a constant of integration.
C
$$\frac{x}{2}(\sin (\log x)+\cos (\log x))+\mathrm{c}$$, where c is a constant of integration.
D
$$-\frac{x}{2}(\sin (\log x)+\cos (\log x))+c$$, where c is a constant of integration.
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