1
JEE Main 2024 (Online) 29th January Morning Shift
+4
-1

For $$x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$, if $$y(x)=\int \frac{\operatorname{cosec} x+\sin x}{\operatorname{cosec} x \sec x+\tan x \sin ^2 x} d x$$, and $$\lim _\limits{x \rightarrow\left(\frac{\pi}{2}\right)^{-}} y(x)=0$$ then $$y\left(\frac{\pi}{4}\right)$$ is equal to

A
$$-\frac{1}{\sqrt{2}} \tan ^{-1}\left(\frac{1}{\sqrt{2}}\right)$$
B
$$\tan ^{-1}\left(\frac{1}{\sqrt{2}}\right)$$
C
$$\frac{1}{2} \tan ^{-1}\left(\frac{1}{\sqrt{2}}\right)$$
D
$$\frac{1}{\sqrt{2}} \tan ^{-1}\left(-\frac{1}{2}\right)$$
2
JEE Main 2024 (Online) 27th January Evening Shift
+4
-1

$$\text { The integral } \int \frac{\left(x^8-x^2\right) \mathrm{d} x}{\left(x^{12}+3 x^6+1\right) \tan ^{-1}\left(x^3+\frac{1}{x^3}\right)} \text { is equal to : }$$

A
$$\log _{\mathrm{e}}\left(\left|\tan ^{-1}\left(x^3+\frac{1}{x^3}\right)\right|\right)^{1 / 3}+\mathrm{C}$$
B
$$\log _{\mathrm{e}}\left(\left|\tan ^{-1}\left(x^3+\frac{1}{x^3}\right)\right|\right)+\mathrm{C}$$
C
$$\log _{\mathrm{e}}\left(\left|\tan ^{-1}\left(x^3+\frac{1}{x^3}\right)\right|\right)^{1 / 2}+\mathrm{C}$$
D
$$\log _{\mathrm{e}}\left(\left|\tan ^{-1}\left(x^3+\frac{1}{x^3}\right)\right|\right)^3+\mathrm{C}$$
3
JEE Main 2023 (Online) 10th April Evening Shift
+4
-1

For $$\alpha, \beta, \gamma, \delta \in \mathbb{N}$$, if $$\int\left(\left(\frac{x}{e}\right)^{2 x}+\left(\frac{e}{x}\right)^{2 x}\right) \log _{e} x d x=\frac{1}{\alpha}\left(\frac{x}{e}\right)^{\beta x}-\frac{1}{\gamma}\left(\frac{e}{x}\right)^{\delta x}+C$$ , where $$e=\sum_\limits{n=0}^{\infty} \frac{1}{n !}$$ and $$\mathrm{C}$$ is constant of integration, then $$\alpha+2 \beta+3 \gamma-4 \delta$$ is equal to :

A
$$-8$$
B
$$-4$$
C
1
D
4

4
JEE Main 2023 (Online) 10th April Morning Shift
+4
-1

If $$I(x) = \int {{e^{{{\sin }^2}x}}(\cos x\sin 2x - \sin x)dx}$$ and $$I(0) = 1$$, then $$I\left( {{\pi \over 3}} \right)$$ is equal to :

A
$$- {e^{{3 \over 4}}}$$
B
$$- {1 \over 2}{e^{{3 \over 4}}}$$
C
$${e^{{3 \over 4}}}$$
D
$${1 \over 2}{e^{{3 \over 4}}}$$
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