MHT CET 2024 15th May Morning Shift | Indefinite Integration Question 14 | Mathematics

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

$2 \tan x$

$\tan x$

$2 \sin x$

$\sin x$

MHT CET 2024 15th May Morning Shift

MCQ (Single Correct Answer)

-0

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

$\frac{1}{24} \log \left(\frac{5+4 \sec x}{5-4 \sec x}\right)+\mathrm{c}$, (where c is a constant of integration)

$\frac{1}{40} \log \left(\frac{5+4 \sec x}{5-4 \sec x}\right)+\mathrm{c}$, (where c is a constant of integration)

$\frac{1}{24} \log \left(\frac{5-4 \sec x}{5+4 \sec x}\right)+\mathrm{c}$, (where c is a constant of integration)

$\frac{1}{40} \log \left(\frac{5-4 \sec x}{5+4 \sec x}\right)+\mathrm{c}$, (where c is a constant of integration)

MHT CET 2024 15th May Morning Shift

MCQ (Single Correct Answer)

-0

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

$\frac{2}{5} \tan ^{-1}\left(\frac{5 \tan \frac{x}{2}+4}{3}\right)+\mathrm{c}$, (where c is a constant of integration)

$\frac{2}{3} \tan ^{-1}\left(\frac{5 \tan \frac{x}{2}+4}{3}\right)+\mathrm{c}$, (where c is a constant of integration)

$\frac{2}{5} \log \left(\frac{5 \tan \frac{x}{2}+7}{5 \tan \frac{x}{2}+1}\right)+\mathrm{c}$, (where c is a constant of integration)

$\frac{2}{3} \log \left(\frac{5 \tan \frac{x}{2}+7}{5 \tan \frac{x}{2}+1}\right)+\mathrm{c}$, (where c is a constant of integration)

MHT CET 2024 11th May Evening Shift

MCQ (Single Correct Answer)

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

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

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

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

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

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