Evaluate the following integrals:

Question:

Evaluate the following integrals:

$\int e^{x}\left(\frac{\sin 4 x-4}{1-\cos 4 x}\right) d x$

Solution:

Let I $=\int e^{x}\left(\frac{\sin 4 x-4}{1-\cos 4 x}\right) d x$

$=\int \mathrm{e}^{\mathrm{x}}\left\{\frac{2 \sin 2 \mathrm{x} \cos 2 \mathrm{x}}{2 \sin ^{2} \mathrm{x}}-\frac{4}{2 \sin ^{2} \mathrm{x}}\right\} \mathrm{dx}$

$=\int \mathrm{e}^{\mathrm{x}}\left\{\cot 2 \mathrm{x}-2 \operatorname{cosec}^{2} 2 \mathrm{x}\right\} \mathrm{dx}$

$\left.=\int \mathrm{e}^{\mathrm{x}} \cot 2 \mathrm{xdx}-\int \mathrm{e}^{\mathrm{x}} 2 \operatorname{cosec}^{2} 2 \mathrm{x}\right\} \mathrm{dx}$

Integrating by parts,

$=e^{x} \cot 2 x-\int e^{x} \frac{d}{d x} \cot 2 x d x-2 \int e^{x} \operatorname{cosec}^{2} 2 x d x$

$=e^{x} \cot 2 x+2 \int e^{x} \operatorname{cosec}^{2} 2 x-2 \int e^{x} \operatorname{cosec}^{2} 2 x$

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

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