Integrate with respect to $x$ .:
$e^{x} sin x$

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Solution

$\int e^{x} sin x d x$

This can be solved by applying the concept of integration by parts

$\int u d v = u v - \int v d u$

Take $sin x$ as $u$ and $e^{x}$ as $d v$ , we get

$\Rightarrow \int e^{x} sin x d x = I$ (let) .... $(1)$

$\Rightarrow I = sin x (e^{x}) - \int cos x e^{x} d x$

Again apply integration by parts

$\Rightarrow I = sin x e^{x} - (cos x e^{x} - \int (- sin x) e^{x} d x)$

$\Rightarrow I = sin x e^{x} - (cos x e^{x} + \int (sin x) e^{x} d x)$

$\Rightarrow I = sin x e^{x} - (cos x e^{x} + I)$ (from $(1)$ )

$\Rightarrow I = sin x e^{x} - cos x e^{x} - I$

$\Rightarrow 2 I = sin x e^{x} - cos x e^{x}$

$\Rightarrow 2 I = e^{x} (sin x - cos x)$

$\Rightarrow I = \frac{e^{x} (sin x - cos x)}{2}$

Therefore, $\Rightarrow \int e^{x} sin x d x = \frac{e^{x} (sin x - cos x)}{2}$

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