Evaluate the following integrals: ^ x ^2x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\text{e}^{\text{x}}\sin^2\text{x }\text{dx}$

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Answer

Let $\text{I}=\int\text{e}^{\text{x}}\sin^2\text{x }\text{dx}$
$=\frac{1}{2}\int\text{e}^\text{x}\ 2\sin^2\text{x dx}$
$=\frac{1}{2}\int\text{e}^\text{x}(1-\cos2\text{x})\text{dx}$
$=\frac{1}{2}\int\text{e}^\text{x}\text{dx}-\frac{1}{2}\int\text{e}^\text{x}\cos2\text{ x dx}$
$\because\ \int\text{e}^{2\text{x}}\cos\text{bx dx}=\frac{\text{e}^{2\text{x}}}{\text{a}^2+\text{b}^2}\{\text{a}\cos\text{bx}-\text{b}\sin\text{bx}\}+\text{C}$
$\therefore\ \text{I}=\frac{1}{2}\big[\text{e}^\text{x}-\frac{\text{e}^\text{x}}{5}\{\cos2\text{x}+2\sin2\text{x}\}\big]+\text{C}$
$\therefore\ \text{I}=\frac{\text{e}^\text{x}}{2}-\frac{\text{e}^\text{x}}{10}\{\cos2\text{x}+2\sin2\text{x}\}+\text{C}$

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