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1
MHT CET 2024 10th May Morning Shift
MCQ (Single Correct Answer)
+2
-0

$$\int \frac{1+\sin (\log x)}{1+\cos (\log x)} d x=$$

A
$x^2 \tan \left(\frac{\log x}{2}\right)+c$, where $c$ is a constant of integration.
B
$x \tan \left(\log \left(\frac{x}{2}\right)\right)+\mathrm{c}$, where c is a constant of integration.
C
$x^3 \log \left(\frac{\tan x}{2}\right)+\mathrm{c}$, where c is a constant of integration.
D
$x \cdot \tan \left(\frac{\log x}{2}\right)+\mathrm{c}$, where c is a constant of integration.
2
MHT CET 2024 9th May Evening Shift
MCQ (Single Correct Answer)
+2
-0

The value of $\int \frac{x+1}{x\left(1+x \mathrm{e}^x\right)^2} \mathrm{dx}$ is equal to

A
$\log \left(\frac{x \mathrm{e}^x}{1+x \mathrm{e}^x}\right)+\frac{x}{1+x \mathrm{e}^x}+\mathrm{c}$, where c is a constant of integration
B
$\log \left(\frac{x \mathrm{e}^x}{1+x \mathrm{e}^x}\right)+\frac{\mathrm{e}^x}{1+x \mathrm{e}^x}+\mathrm{c}$, where c is a constant of integration
C
$\log \left(\frac{x \mathrm{e}^x}{1+x \mathrm{e}^x}\right)+\frac{1}{1+x \mathrm{e}^x}+\mathrm{c}$, where c is a constant of integration
D
$\log \left(\frac{x \mathrm{e}^x}{1+x \mathrm{e}^x}\right)-\frac{x}{1+x \mathrm{e}^x}+\mathrm{c}$, where c is a constant of integration
3
MHT CET 2024 9th May Evening Shift
MCQ (Single Correct Answer)
+2
-0

$$\int \sqrt{\mathrm{e}^x-1} \mathrm{dx}=$$

A
$\sqrt{\mathrm{e}^x-1}+\tan ^{-1} \sqrt{\mathrm{e}^x-1}+\mathrm{c}$, (where c is constant of integration)
B
$2 \sqrt{\mathrm{e}^x-1}+\tan ^{-1} \sqrt{\mathrm{e}^x-1}+\mathrm{c}$, (where c is constant of integration)
C
$2 \sqrt{\mathrm{e}^x-1}-2 \tan ^{-1} \sqrt{\mathrm{e}^x-1}+\mathrm{c}$, (where c is constant of integration)
D
$2 \sqrt{\mathrm{e}^x-1}-\tan ^{-1} \sqrt{\mathrm{e}^x-1}+\mathrm{c}$, (where c is constant of integration)
4
MHT CET 2024 9th May Evening Shift
MCQ (Single Correct Answer)
+2
-0

The value of $\int \frac{\mathrm{d} x}{(x+1)^{3 / 4}(x-2)^{5 / 4}}$ is equal to

A
$4\left(\frac{x+1}{x-2}\right)^{1 / 4}+\mathrm{c}$, where c is a constant of integration.
B
$4\left(\frac{x-2}{x-1}\right)^{1 / 4}+\mathrm{c}$, where c is a constant of integration.
C
$\frac{-4}{3}\left(\frac{x-2}{x+1}\right)^{1 / 4}+\mathrm{c}$, where c is a constant of integration.
D
$\frac{-4}{3}\left(\frac{x+1}{x-2}\right)^{1 / 4}+\mathrm{c}$, where c is a constant of integration.
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