Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Combine fractions.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Multiply $2$ by $- 1$ .

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

Step 4.4

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

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

Step 4.5

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

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify terms.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

Multiply $- 2$ by $- 1$ .

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

Step 7.4

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

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

Step 7.5

Apply the distributive property.

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

Step 7.6

Multiply.

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Step 7.6.1

Multiply $- 1$ by $- 1$ .

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

Step 7.6.2

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

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

Step 7.7

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

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

Step 7.8

Multiply.

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Step 7.8.1

Multiply $2$ by $2$ .

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

Step 7.8.2

Multiply $- 1$ by $- 1$ .

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

Step 7.8.3

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

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

Step 7.9

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

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

Step 7.10

Multiply.

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Step 7.10.1

Multiply $2$ by $- 2$ .

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

Step 7.10.2

Multiply $2$ by $2$ .

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

Step 8

Solving for $\int e^{- 2 x} \sin (2 x) d x$ , we find that $\int e^{- 2 x} \sin (2 x) d x$ = $\frac{- \frac{\sin (2 x) e^{- 2 x}}{2} + \frac{- 2 (\cos (2 x) e^{- 2 x})}{4}}{\frac{8}{4}}$ .

$\frac{- \frac{\sin (2 x) e^{- 2 x}}{2} + \frac{- 2 (\cos (2 x) e^{- 2 x})}{4}}{\frac{8}{4}} + C$

Step 9

Simplify the answer.

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Step 9.1

Simplify.

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Step 9.1.1

Cancel the common factor of $- 2$ and $4$ .

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Step 9.1.1.1

Factor $2$ out of $- 2 \cos (2 x) e^{- 2 x}$ .

$\frac{- \frac{\sin (2 x) e^{- 2 x}}{2} + \frac{2 (- \cos (2 x) e^{- 2 x})}{4}}{\frac{8}{4}} + C$

Step 9.1.1.2

Cancel the common factors.

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Step 9.1.1.2.1

Factor $2$ out of $4$ .

$\frac{- \frac{\sin (2 x) e^{- 2 x}}{2} + \frac{2 (- \cos (2 x) e^{- 2 x})}{2 (2)}}{\frac{8}{4}} + C$

Step 9.1.1.2.2

Cancel the common factor.

$\frac{- \frac{\sin (2 x) e^{- 2 x}}{2} + \frac{2 (- \cos (2 x) e^{- 2 x})}{2 \cdot 2}}{\frac{8}{4}} + C$

Step 9.1.1.2.3

Rewrite the expression.

$\frac{- \frac{\sin (2 x) e^{- 2 x}}{2} + \frac{- \cos (2 x) e^{- 2 x}}{2}}{\frac{8}{4}} + C$

Step 9.1.2

Move the negative in front of the fraction.

$\frac{- \frac{\sin (2 x) e^{- 2 x}}{2} - \frac{\cos (2 x) e^{- 2 x}}{2}}{\frac{8}{4}} + C$

Step 9.1.3

Cancel the common factor of $8$ and $4$ .

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Step 9.1.3.1

Factor $4$ out of $8$ .

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

Step 9.1.3.2

Cancel the common factors.

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Step 9.1.3.2.1

Factor $4$ out of $4$ .

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

Step 9.1.3.2.2

Cancel the common factor.

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

Step 9.1.3.2.3

Rewrite the expression.

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

Step 9.1.3.2.4

Divide $2$ by $1$ .

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

Step 9.2

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

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

Step 9.3

Simplify.

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Step 9.3.1

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

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

Step 9.3.2

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

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

Step 9.3.3

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

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

Step 9.3.4

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

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

Step 9.4

Simplify.

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Step 9.4.1

Apply the distributive property.

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

Step 9.4.2

Multiply $\frac{1}{2} (- \frac{e^{- 2 x} \sin (2 x)}{2})$ .

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Step 9.4.2.1

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

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

Step 9.4.2.2

Multiply $2$ by $2$ .

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

Step 9.4.3

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

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Step 9.4.3.1

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

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

Step 9.4.3.2

Multiply $2$ by $2$ .

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

Step 9.5

Remove parentheses.

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