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

If, $\int \frac{d \theta}{\cos ^2 \theta(\tan 2 \theta+\sec 2 \theta)}=\lambda \tan \theta+2 \log _{\mathrm{e}}|\mathrm{f}(\theta)|+\mathrm{c}$ (where c is a constant of integration), then the ordered pair $(\lambda,|f(\theta)|)$ is equal to

A
$(1,|1+\tan \theta|)$
B
$(1,1-1-\tan \theta \mid)$
C
$(-1,|1+\tan \theta|)$
D
$(-1,|1-\tan \theta|)$
2
MHT CET 2024 10th May Morning Shift
MCQ (Single Correct Answer)
+2
-0

If $\quad \int(2 x+4) \sqrt{x-1} \mathrm{~d} x=\mathrm{a}(x-1)^{\frac{5}{2}}+\mathrm{b}(x-1)^{\frac{3}{2}}+\mathrm{c}$, (where c is a constant of integration), then the value of $a+b$ is

A
$\frac{46}{5}$
B
$\frac{16}{15}$
C
$\frac{24}{5}$
D
$\frac{13}{15}$
3
MHT CET 2024 10th May Morning Shift
MCQ (Single Correct Answer)
+2
-0

$$\int \frac{\sqrt{x}}{x+1} d x=$$

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