1
MHT CET 2024 15th May Morning Shift
MCQ (Single Correct Answer)
+2
-0

If $\int \frac{\mathrm{d} x}{1+3 \sin ^2 x}=\frac{1}{2} \tan ^{-1}(\mathrm{f}(x))+\mathrm{c}$, where c is a constant of integration, then $\mathrm{f}(x)$ is equal to

A
$2 \tan x$
B
$\tan x$
C
$2 \sin x$
D
$\sin x$
2
MHT CET 2024 15th May Morning Shift
MCQ (Single Correct Answer)
+2
-0

The value of $\int \frac{\sec x \cdot \tan x}{9-16 \tan ^2 x} \mathrm{dx}$ is equal to

A
$\frac{1}{24} \log \left(\frac{5+4 \sec x}{5-4 \sec x}\right)+\mathrm{c}$, (where c is a constant of integration)
B
$\frac{1}{40} \log \left(\frac{5+4 \sec x}{5-4 \sec x}\right)+\mathrm{c}$, (where c is a constant of integration)
C
$\frac{1}{24} \log \left(\frac{5-4 \sec x}{5+4 \sec x}\right)+\mathrm{c}$, (where c is a constant of integration)
D
$\frac{1}{40} \log \left(\frac{5-4 \sec x}{5+4 \sec x}\right)+\mathrm{c}$, (where c is a constant of integration)
3
MHT CET 2024 15th May Morning Shift
MCQ (Single Correct Answer)
+2
-0

The value of $\int \frac{d x}{5+4 \sin x}$ is equal to

A
$\frac{2}{5} \tan ^{-1}\left(\frac{5 \tan \frac{x}{2}+4}{3}\right)+\mathrm{c}$, (where c is a constant of integration)
B
$\frac{2}{3} \tan ^{-1}\left(\frac{5 \tan \frac{x}{2}+4}{3}\right)+\mathrm{c}$, (where c is a constant of integration)
C
$\frac{2}{5} \log \left(\frac{5 \tan \frac{x}{2}+7}{5 \tan \frac{x}{2}+1}\right)+\mathrm{c}$, (where c is a constant of integration)
D
$\frac{2}{3} \log \left(\frac{5 \tan \frac{x}{2}+7}{5 \tan \frac{x}{2}+1}\right)+\mathrm{c}$, (where c is a constant of integration)
4
MHT CET 2024 11th May Evening Shift
MCQ (Single Correct Answer)
+2
-0

$$\int \frac{x+1}{x\left(1+x \mathrm{e}^x\right)^2} \mathrm{dx}=$$

A
$\log \left|\frac{x \mathrm{e}^x}{1+x \mathrm{e}^x}\right|+\frac{x}{1+x \mathrm{e}^x}+c$, (where c is a constant of integration)
B
$\quad \log \left|\frac{x \mathrm{e}^x}{1+x \mathrm{e}^x}\right|+\frac{\mathrm{e}^x}{1+x \mathrm{e}^x}+c$, (where c is a constant of integration)
C
$\quad \log \left|\frac{x \mathrm{e}^x}{1+x \mathrm{e}^x}\right|+\frac{1}{1+x \mathrm{e}^x}+c$, (where c is a constant of integration)
D
$\log \left|\frac{x \mathrm{e}^x}{1+x \mathrm{e}^x}\right|-\frac{1}{1+x \mathrm{e}^x}+c$, (where c is a constant of integration)
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