Evaluate the following integrals: ^ x ( 4x-4 1- 4x )dx - Vidyadip

Question

Evaluate the following integrals:

$\int\text{e}^{\text{x}}\Big(\frac{\sin4\text{x}-4}{1-\cos4\text{x}}\Big)\text{dx}$

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Answer

Let $\text{I}=\int\text{e}^{\text{x}}\Big(\frac{\sin4\text{x}-4}{2\sin^22\text{x}}\Big)\text{dx}$
$=\int\text{e}^{\text{x}}\Big\{\frac{2\sin2\text{x}\cos2\text{x}}{2\sin^22\text{x}}-\frac{4}{2\sin^{2}2\text{x}}\Big\}\text{dx}$
$=\int\text{e}^{\text{x}}\big(\cot2\text{x}-2\text{cosec}^22\text{x}\big)\text{dx}$
$=\int\text{e}^{\text{x}}\cot2\text{x dx}-2\int\text{e}^{\text{x}}\text{cosec}^22\text{x dx}$
integrating by parts
$=\text{e}^{\text{x}}\cot2\text{x}-\int\text{e}^{\text{x}}\frac{\text{d}}{\text{dx}}(\cot2\text{x})\text{dx}-2\int\text{e}^{\text{x}}\text{cosec}^22\text{x dx}$
$=\text{e}^{\text{x}}\cot2\text{x}+2\int\text{e}^{\text{x}}\text{cosec}^22\text{x}-2\int\text{e}^{\text{x}}\text{cosec}^22\text{x dx}$
$=\text{e}^{\text{x}}\cot2\text{x}+\text{C}$

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