Calculus Examples

$\int e^{x} \sin (4 x) d x$

Step 1

Reorder $e^{x}$ and $\sin (4 x)$ .

$\int \sin (4 x) e^{x} d x$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \sin (4 x)$ and $d v = e^{x}$ .

$\sin (4 x) e^{x} - \int e^{x} (4 \cos (4 x)) d x$

Step 3

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$\sin (4 x) e^{x} - (4 \int e^{x} (\cos (4 x)) d x)$

Step 4

Simplify the expression.

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Step 4.1

Multiply $4$ by $- 1$ .

$\sin (4 x) e^{x} - 4 \int e^{x} (\cos (4 x)) d x$

Step 4.2

Reorder $e^{x}$ and $\cos (4 x)$ .

$\sin (4 x) e^{x} - 4 \int \cos (4 x) e^{x} d x$

Step 5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \cos (4 x)$ and $d v = e^{x}$ .

$\sin (4 x) e^{x} - 4 (\cos (4 x) e^{x} - \int e^{x} (- 4 \sin (4 x)) d x)$

Step 6

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$\sin (4 x) e^{x} - 4 (\cos (4 x) e^{x} - (- 4 \int e^{x} (\sin (4 x)) d x))$

Step 7

Simplify by multiplying through.

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Step 7.1

Multiply $- 4$ by $- 1$ .

$\sin (4 x) e^{x} - 4 (\cos (4 x) e^{x} + 4 \int e^{x} (\sin (4 x)) d x)$

Step 7.2

Apply the distributive property.

$\sin (4 x) e^{x} - 4 (\cos (4 x) e^{x}) - 4 (4 \int e^{x} (\sin (4 x)) d x)$

Step 7.3

Multiply $4$ by $- 4$ .

$\sin (4 x) e^{x} - 4 (\cos (4 x) e^{x}) - 16 \int e^{x} (\sin (4 x)) d x$

Step 8

Solving for $\int e^{x} \sin (4 x) d x$ , we find that $\int e^{x} \sin (4 x) d x$ = $\frac{\sin (4 x) e^{x} - 4 (\cos (4 x) e^{x})}{17}$ .

$\frac{\sin (4 x) e^{x} - 4 (\cos (4 x) e^{x})}{17} + C$

Step 9

Rewrite $\frac{\sin (4 x) e^{x} - 4 \cos (4 x) e^{x}}{17} + C$ as $\frac{1}{17} (\sin (4 x) e^{x} - 4 \cos (4 x) e^{x}) + C$ .

$\frac{1}{17} (\sin (4 x) e^{x} - 4 \cos (4 x) e^{x}) + C$