$\int \frac{xcos x + 1}{\sqrt{2 x^{3} e^{\sin x} + x^{2}}} dx$

Moderate

A

$\log |\frac{\sqrt{2 {xe}^{\sin x} + 1} - 1}{\sqrt{2 {xe}^{\sin x} + 1} + 1}| + C$
B

$\log |\frac{\sqrt{2 {xe}^{\sin x} - 1} + 1}{\sqrt{2 {xe}^{\sin x} + 1} + 1}| + C$
C

$\log |\frac{\sqrt{2 {xe}^{\sin x} + 1} + 1}{\sqrt{2 {xe}^{\sin x} - 1} + 1}| + C$
D

$\log |\frac{\sqrt{2 {xe}^{\sin x} + 1} + 1}{\sqrt{2 {xe}^{\sin x} - 1} - 1}| + C$

Solution

$I = \int \frac{(xcos x + 1) e^{\sin x} dx}{e^{\sin x} x \sqrt{2 {xe}^{\sin x} + 1}}$
$Put 2 {xe}^{\sin x} + 1 = t^{2}$
$\begin{aligned} \Rightarrow 2 [e^{\sin x} + e^{\sin x} \cos xx] dx = 2 tdt \\ \Rightarrow e^{\sin x} (1 + xcos x) dx = tdt \\ ∴ I = \int \frac{tdt}{\frac{(t^{2} - 1)}{2} \cdot t} \\ I = 2 \int \frac{dt}{t^{2} - 1} = \log |\frac{t - 1}{t + 1}| + C, \end{aligned}$
where $t^{2} = 2 {xe}^{\sin x} + 1$

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