Calculus Examples

$\int e^{3 x} \cos (2 x) d x$

Step 1

Reorder $e^{3 x}$ and $\cos (2 x)$ .

$\int \cos (2 x) e^{3 x} d x$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \cos (2 x)$ and $d v = e^{3 x}$ .

$\cos (2 x) (\frac{1}{3} e^{3 x}) - \int \frac{1}{3} e^{3 x} (- 2 \sin (2 x)) d x$

Step 3

Since $\frac{1}{3} \cdot - 2$ is constant with respect to $x$ , move $\frac{1}{3} \cdot - 2$ out of the integral.

$\cos (2 x) (\frac{1}{3} e^{3 x}) - (\frac{1}{3} \cdot - 2 \int e^{3 x} (\sin (2 x)) d x)$

Step 4

Combine fractions.

Tap for more steps...

Step 4.1

Combine $\frac{1}{3}$ and $e^{3 x}$ .

$\cos (2 x) \frac{e^{3 x}}{3} - (\frac{1}{3} \cdot - 2 \int e^{3 x} (\sin (2 x)) d x)$

Step 4.2

Combine $\cos (2 x)$ and $\frac{e^{3 x}}{3}$ .

$\frac{\cos (2 x) e^{3 x}}{3} - (\frac{1}{3} \cdot - 2 \int e^{3 x} (\sin (2 x)) d x)$

Step 4.3

Combine $\frac{1}{3}$ and $- 2$ .

$\frac{\cos (2 x) e^{3 x}}{3} - (\frac{- 2}{3} \int e^{3 x} (\sin (2 x)) d x)$

Step 4.4

Reorder $e^{3 x}$ and $\sin (2 x)$ .

$\frac{\cos (2 x) e^{3 x}}{3} - (\frac{- 2}{3} \int \sin (2 x) e^{3 x} d x)$

Step 5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \sin (2 x)$ and $d v = e^{3 x}$ .

$\frac{\cos (2 x) e^{3 x}}{3} - (\frac{- 2}{3} (\sin (2 x) (\frac{1}{3} e^{3 x}) - \int \frac{1}{3} e^{3 x} (2 \cos (2 x)) d x))$

Step 6

Since $\frac{1}{3} \cdot 2$ is constant with respect to $x$ , move $\frac{1}{3} \cdot 2$ out of the integral.

$\frac{\cos (2 x) e^{3 x}}{3} - (\frac{- 2}{3} (\sin (2 x) (\frac{1}{3} e^{3 x}) - (\frac{1}{3} \cdot 2 \int e^{3 x} (\cos (2 x)) d x)))$

Step 7

Simplify terms.

Tap for more steps...

Step 7.1

Combine $\frac{1}{3}$ and $e^{3 x}$ .

$\frac{\cos (2 x) e^{3 x}}{3} - (\frac{- 2}{3} (\sin (2 x) \frac{e^{3 x}}{3} - (\frac{1}{3} \cdot 2 \int e^{3 x} (\cos (2 x)) d x)))$

Step 7.2

Combine $\sin (2 x)$ and $\frac{e^{3 x}}{3}$ .

$\frac{\cos (2 x) e^{3 x}}{3} - (\frac{- 2}{3} (\frac{\sin (2 x) e^{3 x}}{3} - (\frac{1}{3} \cdot 2 \int e^{3 x} (\cos (2 x)) d x)))$

Step 7.3

Combine $\frac{1}{3}$ and $2$ .

$\frac{\cos (2 x) e^{3 x}}{3} - (\frac{- 2}{3} (\frac{\sin (2 x) e^{3 x}}{3} - (\frac{2}{3} \int e^{3 x} (\cos (2 x)) d x)))$

Step 7.4

Apply the distributive property.

$\frac{\cos (2 x) e^{3 x}}{3} - \frac{- 2}{3} \cdot \frac{\sin (2 x) e^{3 x}}{3} - \frac{- 2}{3} (- (\frac{2}{3} \int e^{3 x} (\cos (2 x)) d x))$

Step 7.5

Multiply $\frac{\sin (2 x) e^{3 x}}{3}$ by $\frac{- 2}{3}$ .

$\frac{\cos (2 x) e^{3 x}}{3} - \frac{\sin (2 x) e^{3 x} \cdot - 2}{3 \cdot 3} - \frac{- 2}{3} (- (\frac{2}{3} \int e^{3 x} (\cos (2 x)) d x))$

Step 7.6

Multiply.

Tap for more steps...

Step 7.6.1

Multiply $3$ by $3$ .

$\frac{\cos (2 x) e^{3 x}}{3} - \frac{\sin (2 x) e^{3 x} \cdot - 2}{9} - \frac{- 2}{3} (- (\frac{2}{3} \int e^{3 x} (\cos (2 x)) d x))$

Step 7.6.2

Multiply $- 1$ by $- 1$ .

$\frac{\cos (2 x) e^{3 x}}{3} - \frac{\sin (2 x) e^{3 x} \cdot - 2}{9} + 1 (\frac{- 2}{3}) (\frac{2}{3} \int e^{3 x} (\cos (2 x)) d x)$

Step 7.6.3

Multiply $\frac{- 2}{3}$ by $1$ .

$\frac{\cos (2 x) e^{3 x}}{3} - \frac{\sin (2 x) e^{3 x} \cdot - 2}{9} + \frac{- 2}{3} (\frac{2}{3} \int e^{3 x} (\cos (2 x)) d x)$

Step 7.7

Multiply $\frac{2}{3}$ by $\frac{- 2}{3}$ .

$\frac{\cos (2 x) e^{3 x}}{3} - \frac{\sin (2 x) e^{3 x} \cdot - 2}{9} + \frac{2 \cdot - 2}{3 \cdot 3} \int e^{3 x} (\cos (2 x)) d x$

Step 7.8

Multiply.

Tap for more steps...

Step 7.8.1

Multiply $2$ by $- 2$ .

$\frac{\cos (2 x) e^{3 x}}{3} - \frac{\sin (2 x) e^{3 x} \cdot - 2}{9} + \frac{- 4}{3 \cdot 3} \int e^{3 x} (\cos (2 x)) d x$

Step 7.8.2

Multiply $3$ by $3$ .

$\frac{\cos (2 x) e^{3 x}}{3} - \frac{\sin (2 x) e^{3 x} \cdot - 2}{9} + \frac{- 4}{9} \int e^{3 x} (\cos (2 x)) d x$

Step 8

Solving for $\int e^{3 x} \cos (2 x) d x$ , we find that $\int e^{3 x} \cos (2 x) d x$ = $\frac{\frac{\cos (2 x) e^{3 x}}{3} - \frac{\sin (2 x) e^{3 x} \cdot - 2}{9}}{\frac{13}{9}}$ .

$\frac{\frac{\cos (2 x) e^{3 x}}{3} - \frac{\sin (2 x) e^{3 x} \cdot - 2}{9}}{\frac{13}{9}} + C$

Step 9

Simplify the answer.

Tap for more steps...

Step 9.1

Simplify.

Tap for more steps...

Step 9.1.1

Move $- 2$ to the left of $\sin (2 x) e^{3 x}$ .

$\frac{\frac{\cos (2 x) e^{3 x}}{3} - \frac{- 2 \cdot (\sin (2 x) e^{3 x})}{9}}{\frac{13}{9}} + C$

Step 9.1.2

Move the negative in front of the fraction.

$\frac{\frac{\cos (2 x) e^{3 x}}{3} - - \frac{2 \sin (2 x) e^{3 x}}{9}}{\frac{13}{9}} + C$

Step 9.1.3

Multiply $- 1$ by $- 1$ .

$\frac{\frac{\cos (2 x) e^{3 x}}{3} + 1 \frac{2 \sin (2 x) e^{3 x}}{9}}{\frac{13}{9}} + C$

Step 9.1.4

Multiply $\frac{2 \sin (2 x) e^{3 x}}{9}$ by $1$ .

$\frac{\frac{\cos (2 x) e^{3 x}}{3} + \frac{2 \sin (2 x) e^{3 x}}{9}}{\frac{13}{9}} + C$

Step 9.1.5

Multiply by the reciprocal of the fraction to divide by $\frac{13}{9}$ .

$(\frac{\cos (2 x) e^{3 x}}{3} + \frac{2 \sin (2 x) e^{3 x}}{9}) \frac{9}{13} + C$

Step 9.2

Rewrite $(\frac{\cos (2 x) e^{3 x}}{3} + \frac{2 \sin (2 x) e^{3 x}}{9}) \frac{9}{13} + C$ as $(\frac{1}{3} \cos (2 x) e^{3 x} + \frac{2}{9} \sin (2 x) e^{3 x}) \frac{9}{13} + C$ .

$(\frac{1}{3} \cos (2 x) e^{3 x} + \frac{2}{9} \sin (2 x) e^{3 x}) \frac{9}{13} + C$