Calculus Examples

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

Step 1

Reorder $e^{x}$ and $\sin (x)$ .

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

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \sin (x)$ and $d v = e^{x}$ .

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

Step 3

Reorder $e^{x}$ and $\cos (x)$ .

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

Step 4

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \cos (x)$ and $d v = e^{x}$ .

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

Step 5

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 6

Simplify by multiplying through.

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Step 6.1

Multiply $- 1$ by $- 1$ .

$\sin (x) e^{x} - (\cos (x) e^{x} + 1 \int e^{x} (\sin (x)) d x)$

Step 6.2

Multiply $\int e^{x} (\sin (x)) d x$ by $1$ .

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

Step 6.3

Apply the distributive property.

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

Step 7

Solving for $\int e^{x} \sin (x) d x$ , we find that $\int e^{x} \sin (x) d x$ = $\frac{\sin (x) e^{x} - (\cos (x) e^{x})}{2}$ .

$\frac{\sin (x) e^{x} - (\cos (x) e^{x})}{2} + C$

Step 8

Rewrite $\frac{\sin (x) e^{x} - \cos (x) e^{x}}{2} + C$ as $\frac{1}{2} (\sin (x) e^{x} - \cos (x) e^{x}) + C$ .

$\frac{1}{2} (\sin (x) e^{x} - \cos (x) e^{x}) + C$