1
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

2
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
$${1 \over 2}{e^{{3 \over 4}}}$$
3
JEE Main 2023 (Online) 8th April Evening Shift
+4
-1

The integral $$\int\left[\left(\frac{x}{2}\right)^x+\left(\frac{2}{x}\right)^x\right] \ln \left(\frac{e x}{2}\right) d x$$ is equal to :

A
$$\left(\frac{x}{2}\right)^{x}+\left(\frac{2}{x}\right)^{x}+C$$
B
$$\left(\frac{x}{2}\right)^{x}-\left(\frac{2}{x}\right)^{x}+C$$
C
$$\left(\frac{x}{2}\right)^{x} \log _{2}\left(\frac{2}{x}\right)+C$$
D
None
4
JEE Main 2023 (Online) 8th April Morning Shift
+4
-1

Let $$I(x)=\int \frac{(x+1)}{x\left(1+x e^{x}\right)^{2}} d x, x > 0$$. If $$\lim_\limits{x \rightarrow \infty} I(x)=0$$, then $$I(1)$$ is equal to :

A
$$\frac{e+1}{e+2}-\log _{e}(e+1)$$
B
$$\frac{e+1}{e+2}+\log _{e}(e+1)$$
C
$$\frac{e+2}{e+1}-\log _{e}(e+1)$$
D
$$\frac{e+2}{e+1}+\log _{e}(e+1)$$
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