Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Combine fractions.

Tap for more steps...

Step 4.1

Combine $\frac{1}{2}$ and $e^{2 x}$ .

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify terms.

Tap for more steps...

Step 7.1

Combine $\frac{1}{2}$ and $e^{2 x}$ .

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

Apply the distributive property.

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

Step 7.5

Multiply $\frac{\cos (x) e^{2 x}}{2}$ by $\frac{1}{2}$ .

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

Step 7.6

Multiply.

Tap for more steps...

Step 7.6.1

Multiply $2$ by $2$ .

$\frac{\sin (x) e^{2 x}}{2} - \frac{\cos (x) e^{2 x}}{4} - \frac{1}{2} (- (\frac{- 1}{2} \int e^{2 x} (\sin (x)) d x))$

Step 7.6.2

Multiply $- 1$ by $- 1$ .

$\frac{\sin (x) e^{2 x}}{2} - \frac{\cos (x) e^{2 x}}{4} + 1 (\frac{1}{2}) (\frac{- 1}{2} \int e^{2 x} (\sin (x)) d x)$

Step 7.6.3

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

$\frac{\sin (x) e^{2 x}}{2} - \frac{\cos (x) e^{2 x}}{4} + \frac{1}{2} (\frac{- 1}{2} \int e^{2 x} (\sin (x)) d x)$

Step 7.7

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

$\frac{\sin (x) e^{2 x}}{2} - \frac{\cos (x) e^{2 x}}{4} + \frac{- 1}{2 \cdot 2} \int e^{2 x} (\sin (x)) d x$

Step 7.8

Multiply $2$ by $2$ .

$\frac{\sin (x) e^{2 x}}{2} - \frac{\cos (x) e^{2 x}}{4} + \frac{- 1}{4} \int e^{2 x} (\sin (x)) d x$

Step 8

Solving for $\int e^{2 x} \sin (x) d x$ , we find that $\int e^{2 x} \sin (x) d x$ = $\frac{\frac{\sin (x) e^{2 x}}{2} - \frac{\cos (x) e^{2 x}}{4}}{\frac{5}{4}}$ .

$\frac{\frac{\sin (x) e^{2 x}}{2} - \frac{\cos (x) e^{2 x}}{4}}{\frac{5}{4}} + C$

Step 9

Simplify the answer.

Tap for more steps...

Step 9.1

Simplify the answer.

Tap for more steps...

Step 9.1.1

Multiply by the reciprocal of the fraction to divide by $\frac{5}{4}$ .

$(\frac{\sin (x) e^{2 x}}{2} - \frac{\cos (x) e^{2 x}}{4}) \frac{4}{5} + C$

Step 9.1.2

Rewrite $(\frac{\sin (x) e^{2 x}}{2} - \frac{\cos (x) e^{2 x}}{4}) \frac{4}{5} + C$ as $(\frac{1}{2} \sin (x) e^{2 x} - \frac{1}{4} \cos (x) e^{2 x}) \frac{4}{5} + C$ .

$(\frac{1}{2} \sin (x) e^{2 x} - \frac{1}{4} \cos (x) e^{2 x}) \frac{4}{5} + C$

Step 9.1.3

Simplify.

Tap for more steps...

Step 9.1.3.1

Combine $\frac{1}{2}$ and $\sin (x)$ .

$(\frac{\sin (x)}{2} e^{2 x} - \frac{1}{4} \cos (x) e^{2 x}) \frac{4}{5} + C$

Step 9.1.3.2

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

$(\frac{\sin (x) e^{2 x}}{2} - \frac{1}{4} \cos (x) e^{2 x}) \frac{4}{5} + C$

Step 9.1.3.3

Combine $\cos (x)$ and $\frac{1}{4}$ .

$(\frac{\sin (x) e^{2 x}}{2} - \frac{\cos (x)}{4} e^{2 x}) \frac{4}{5} + C$

Step 9.1.3.4

Combine $e^{2 x}$ and $\frac{\cos (x)}{4}$ .

$(\frac{\sin (x) e^{2 x}}{2} - \frac{e^{2 x} \cos (x)}{4}) \frac{4}{5} + C$

Step 9.1.4

Reorder factors in $\frac{\sin (x) e^{2 x}}{2}$ .

$(\frac{1}{2} e^{2 x} \sin (x) - \frac{1}{4} e^{2 x} \cos (x)) \frac{4}{5} + C$

Step 9.2

Rewrite $(\frac{1}{2} e^{2 x} \sin (x) - \frac{1}{4} e^{2 x} \cos (x)) \frac{4}{5} + C$ as $(\frac{e^{2 x} \sin (x)}{2} - \frac{\cos (x) e^{2 x}}{4}) \frac{4}{5} + C$ .

$(\frac{e^{2 x} \sin (x)}{2} - \frac{\cos (x) e^{2 x}}{4}) \frac{4}{5} + C$

Step 9.3

Reorder factors in $\frac{\cos (x) e^{2 x}}{4}$ .

$(\frac{1}{2} e^{2 x} \sin (x) - \frac{1}{4} e^{2 x} \cos (x)) \frac{4}{5} + C$