Calculus Examples

$\int e^{x} \cos (2 x) d x$

Step 1

Reorder $e^{x}$ and $\cos (2 x)$ .

$\int \cos (2 x) e^{x} d x$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \cos (2 x)$ and $d v = e^{x}$ .

$\cos (2 x) e^{x} - \int e^{x} (- 2 \sin (2 x)) d x$

Step 3

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

$\cos (2 x) e^{x} - (- 2 \int e^{x} (\sin (2 x)) d x)$

Step 4

Simplify the expression.

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Step 4.1

Multiply $- 2$ by $- 1$ .

$\cos (2 x) e^{x} + 2 \int e^{x} (\sin (2 x)) d x$

Step 4.2

Reorder $e^{x}$ and $\sin (2 x)$ .

$\cos (2 x) e^{x} + 2 \int \sin (2 x) e^{x} d x$

Step 5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \sin (2 x)$ and $d v = e^{x}$ .

$\cos (2 x) e^{x} + 2 (\sin (2 x) e^{x} - \int e^{x} (2 \cos (2 x)) d x)$

Step 6

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$\cos (2 x) e^{x} + 2 (\sin (2 x) e^{x} - (2 \int e^{x} (\cos (2 x)) d x))$

Step 7

Simplify by multiplying through.

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Step 7.1

Multiply $2$ by $- 1$ .

$\cos (2 x) e^{x} + 2 (\sin (2 x) e^{x} - 2 \int e^{x} (\cos (2 x)) d x)$

Step 7.2

Apply the distributive property.

$\cos (2 x) e^{x} + 2 (\sin (2 x) e^{x}) + 2 (- 2 \int e^{x} (\cos (2 x)) d x)$

Step 7.3

Multiply $- 2$ by $2$ .

$\cos (2 x) e^{x} + 2 (\sin (2 x) e^{x}) - 4 \int e^{x} (\cos (2 x)) d x$

Step 8

Solving for $\int e^{x} \cos (2 x) d x$ , we find that $\int e^{x} \cos (2 x) d x$ = $\frac{\cos (2 x) e^{x} + 2 (\sin (2 x) e^{x})}{5}$ .

$\frac{\cos (2 x) e^{x} + 2 (\sin (2 x) e^{x})}{5} + C$

Step 9

Rewrite $\frac{\cos (2 x) e^{x} + 2 \sin (2 x) e^{x}}{5} + C$ as $\frac{1}{5} (\cos (2 x) e^{x} + 2 \sin (2 x) e^{x}) + C$ .

$\frac{1}{5} (\cos (2 x) e^{x} + 2 \sin (2 x) e^{x}) + C$