Mark the correct alternative in each of the following:

Question:

Mark the correct alternative in each of the following:

Evaluate $\int \mathrm{e}^{\mathrm{x}}\left(\frac{1-\sin \mathrm{x}}{1-\cos \mathrm{x}}\right) \mathrm{dx}$

A. $-\mathrm{e}^{\mathrm{x}} \tan \frac{\mathrm{x}}{2}+\mathrm{C}$

B. $-e^{x} \cot \frac{x}{2}+C$

C. $-\frac{1}{2} \mathrm{e}^{\mathrm{x}} \tan \frac{\mathrm{x}}{2}+\mathrm{C}$

D. $-\frac{1}{2} e^{x} \cot \frac{x}{2}+C$

Solution:

Given, $\int e^{x}\left(\frac{1-\sin x}{1-\cos x}\right) d x$

$=-\int e^{x}\left(\frac{\sin x}{1-\cos x}-\frac{1}{1-\cos x}\right) d x\left\{\int e^{x}\left[f(x)+f^{\prime}(x)\right]=e^{x} f(x)\right\}$

$\Rightarrow f(x)=\frac{\sin x}{1-\cos x} ; f^{\prime}(x)=-\frac{1}{1-\cos x}$

$=-e^{x}\left(\frac{\sin x}{1-\cos x}\right)$

$\because\left[\frac{\sin x}{1-\cos x}=\cot \frac{x}{2}\right]$

$=-e^{x} \cot \frac{x}{2}+c$

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