Calculus Examples

$\int e^{- 3 x} \sin (5 x) d x$

Step 1

Reorder $e^{- 3 x}$ and $\sin (5 x)$ .

$\int \sin (5 x) e^{- 3 x} d x$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \sin (5 x)$ and $d v = e^{- 3 x}$ .

$\sin (5 x) (- \frac{1}{3} e^{- 3 x}) - \int - \frac{1}{3} e^{- 3 x} (5 \cos (5 x)) d x$

Step 3

Since $- \frac{1}{3} \cdot 5$ is constant with respect to $x$ , move $- \frac{1}{3} \cdot 5$ out of the integral.

$\sin (5 x) (- \frac{1}{3} e^{- 3 x}) - (- \frac{1}{3} \cdot 5 \int e^{- 3 x} (\cos (5 x)) d x)$

Step 4

Combine fractions.

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Step 4.1

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

$\sin (5 x) (- \frac{e^{- 3 x}}{3}) - (- \frac{1}{3} \cdot 5 \int e^{- 3 x} (\cos (5 x)) d x)$

Step 4.2

Combine $\sin (5 x)$ and $\frac{e^{- 3 x}}{3}$ .

$- \frac{\sin (5 x) e^{- 3 x}}{3} - (- \frac{1}{3} \cdot 5 \int e^{- 3 x} (\cos (5 x)) d x)$

Step 4.3

Multiply $5$ by $- 1$ .

$- \frac{\sin (5 x) e^{- 3 x}}{3} - (- 5 (\frac{1}{3}) \int e^{- 3 x} (\cos (5 x)) d x)$

Step 4.4

Combine $- 5$ and $\frac{1}{3}$ .

$- \frac{\sin (5 x) e^{- 3 x}}{3} - (\frac{- 5}{3} \int e^{- 3 x} (\cos (5 x)) d x)$

Step 4.5

Reorder $e^{- 3 x}$ and $\cos (5 x)$ .

$- \frac{\sin (5 x) e^{- 3 x}}{3} - (\frac{- 5}{3} \int \cos (5 x) e^{- 3 x} d x)$

Step 5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \cos (5 x)$ and $d v = e^{- 3 x}$ .

$- \frac{\sin (5 x) e^{- 3 x}}{3} - (\frac{- 5}{3} (\cos (5 x) (- \frac{1}{3} e^{- 3 x}) - \int - \frac{1}{3} e^{- 3 x} (- 5 \sin (5 x)) d x))$

Step 6

Since $- \frac{1}{3} \cdot - 5$ is constant with respect to $x$ , move $- \frac{1}{3} \cdot - 5$ out of the integral.

$- \frac{\sin (5 x) e^{- 3 x}}{3} - (\frac{- 5}{3} (\cos (5 x) (- \frac{1}{3} e^{- 3 x}) - (- \frac{1}{3} \cdot - 5 \int e^{- 3 x} (\sin (5 x)) d x)))$

Step 7

Simplify terms.

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Step 7.1

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

$- \frac{\sin (5 x) e^{- 3 x}}{3} - (\frac{- 5}{3} (\cos (5 x) (- \frac{e^{- 3 x}}{3}) - (- \frac{1}{3} \cdot - 5 \int e^{- 3 x} (\sin (5 x)) d x)))$

Step 7.2

Combine $\cos (5 x)$ and $\frac{e^{- 3 x}}{3}$ .

$- \frac{\sin (5 x) e^{- 3 x}}{3} - (\frac{- 5}{3} (- \frac{\cos (5 x) e^{- 3 x}}{3} - (- \frac{1}{3} \cdot - 5 \int e^{- 3 x} (\sin (5 x)) d x)))$

Step 7.3

Multiply $- 5$ by $- 1$ .

$- \frac{\sin (5 x) e^{- 3 x}}{3} - (\frac{- 5}{3} (- \frac{\cos (5 x) e^{- 3 x}}{3} - (5 (\frac{1}{3}) \int e^{- 3 x} (\sin (5 x)) d x)))$

Step 7.4

Combine $5$ and $\frac{1}{3}$ .

$- \frac{\sin (5 x) e^{- 3 x}}{3} - (\frac{- 5}{3} (- \frac{\cos (5 x) e^{- 3 x}}{3} - (\frac{5}{3} \int e^{- 3 x} (\sin (5 x)) d x)))$

Step 7.5

Apply the distributive property.

$- \frac{\sin (5 x) e^{- 3 x}}{3} - \frac{- 5}{3} (- \frac{\cos (5 x) e^{- 3 x}}{3}) - \frac{- 5}{3} (- (\frac{5}{3} \int e^{- 3 x} (\sin (5 x)) d x))$

Step 7.6

Multiply.

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Step 7.6.1

Multiply $- 1$ by $- 1$ .

$- \frac{\sin (5 x) e^{- 3 x}}{3} + 1 (\frac{- 5}{3}) \frac{\cos (5 x) e^{- 3 x}}{3} - \frac{- 5}{3} (- (\frac{5}{3} \int e^{- 3 x} (\sin (5 x)) d x))$

Step 7.6.2

Multiply $\frac{- 5}{3}$ by $1$ .

$- \frac{\sin (5 x) e^{- 3 x}}{3} + \frac{- 5}{3} \cdot \frac{\cos (5 x) e^{- 3 x}}{3} - \frac{- 5}{3} (- (\frac{5}{3} \int e^{- 3 x} (\sin (5 x)) d x))$

Step 7.7

Multiply $\frac{- 5}{3}$ by $\frac{\cos (5 x) e^{- 3 x}}{3}$ .

$- \frac{\sin (5 x) e^{- 3 x}}{3} + \frac{- 5 (\cos (5 x) e^{- 3 x})}{3 \cdot 3} - \frac{- 5}{3} (- (\frac{5}{3} \int e^{- 3 x} (\sin (5 x)) d x))$

Step 7.8

Multiply.

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Step 7.8.1

Multiply $3$ by $3$ .

$- \frac{\sin (5 x) e^{- 3 x}}{3} + \frac{- 5 (\cos (5 x) e^{- 3 x})}{9} - \frac{- 5}{3} (- (\frac{5}{3} \int e^{- 3 x} (\sin (5 x)) d x))$

Step 7.8.2

Multiply $- 1$ by $- 1$ .

$- \frac{\sin (5 x) e^{- 3 x}}{3} + \frac{- 5 (\cos (5 x) e^{- 3 x})}{9} + 1 (\frac{- 5}{3}) (\frac{5}{3} \int e^{- 3 x} (\sin (5 x)) d x)$

Step 7.8.3

Multiply $\frac{- 5}{3}$ by $1$ .

$- \frac{\sin (5 x) e^{- 3 x}}{3} + \frac{- 5 (\cos (5 x) e^{- 3 x})}{9} + \frac{- 5}{3} (\frac{5}{3} \int e^{- 3 x} (\sin (5 x)) d x)$

Step 7.9

Multiply $\frac{5}{3}$ by $\frac{- 5}{3}$ .

$- \frac{\sin (5 x) e^{- 3 x}}{3} + \frac{- 5 (\cos (5 x) e^{- 3 x})}{9} + \frac{5 \cdot - 5}{3 \cdot 3} \int e^{- 3 x} (\sin (5 x)) d x$

Step 7.10

Multiply.

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Step 7.10.1

Multiply $5$ by $- 5$ .

$- \frac{\sin (5 x) e^{- 3 x}}{3} + \frac{- 5 (\cos (5 x) e^{- 3 x})}{9} + \frac{- 25}{3 \cdot 3} \int e^{- 3 x} (\sin (5 x)) d x$

Step 7.10.2

Multiply $3$ by $3$ .

$- \frac{\sin (5 x) e^{- 3 x}}{3} + \frac{- 5 (\cos (5 x) e^{- 3 x})}{9} + \frac{- 25}{9} \int e^{- 3 x} (\sin (5 x)) d x$

Step 8

Solving for $\int e^{- 3 x} \sin (5 x) d x$ , we find that $\int e^{- 3 x} \sin (5 x) d x$ = $\frac{- \frac{\sin (5 x) e^{- 3 x}}{3} + \frac{- 5 (\cos (5 x) e^{- 3 x})}{9}}{\frac{34}{9}}$ .

$\frac{- \frac{\sin (5 x) e^{- 3 x}}{3} + \frac{- 5 (\cos (5 x) e^{- 3 x})}{9}}{\frac{34}{9}} + C$

Step 9

Simplify the answer.

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Step 9.1

Simplify.

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Step 9.1.1

Move the negative in front of the fraction.

$\frac{- \frac{\sin (5 x) e^{- 3 x}}{3} - \frac{5 \cos (5 x) e^{- 3 x}}{9}}{\frac{34}{9}} + C$

Step 9.1.2

Multiply by the reciprocal of the fraction to divide by $\frac{34}{9}$ .

$(- \frac{\sin (5 x) e^{- 3 x}}{3} - \frac{5 \cos (5 x) e^{- 3 x}}{9}) \frac{9}{34} + C$

Step 9.2

Rewrite $(- \frac{\sin (5 x) e^{- 3 x}}{3} - \frac{5 \cos (5 x) e^{- 3 x}}{9}) \frac{9}{34} + C$ as $(- \frac{1}{3} \sin (5 x) e^{- 3 x} - \frac{5}{9} \cos (5 x) e^{- 3 x}) \frac{9}{34} + C$ .

$(- \frac{1}{3} \sin (5 x) e^{- 3 x} - \frac{5}{9} \cos (5 x) e^{- 3 x}) \frac{9}{34} + C$

Step 9.3

Simplify.

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Step 9.3.1

Combine $\sin (5 x)$ and $\frac{1}{3}$ .

$(- \frac{\sin (5 x)}{3} e^{- 3 x} - \frac{5}{9} \cos (5 x) e^{- 3 x}) \frac{9}{34} + C$

Step 9.3.2

Combine $e^{- 3 x}$ and $\frac{\sin (5 x)}{3}$ .

$(- \frac{e^{- 3 x} \sin (5 x)}{3} - \frac{5}{9} \cos (5 x) e^{- 3 x}) \frac{9}{34} + C$

Step 9.3.3

Combine $\cos (5 x)$ and $\frac{5}{9}$ .

$(- \frac{e^{- 3 x} \sin (5 x)}{3} - \frac{\cos (5 x) \cdot 5}{9} e^{- 3 x}) \frac{9}{34} + C$

Step 9.3.4

Combine $e^{- 3 x}$ and $\frac{\cos (5 x) \cdot 5}{9}$ .

$(- \frac{e^{- 3 x} \sin (5 x)}{3} - \frac{e^{- 3 x} (\cos (5 x) \cdot 5)}{9}) \frac{9}{34} + C$

Step 9.3.5

Move $5$ to the left of $\cos (5 x)$ .

$(- \frac{e^{- 3 x} \sin (5 x)}{3} - \frac{e^{- 3 x} (5 \cos (5 x))}{9}) \frac{9}{34} + C$

Step 9.4

Simplify.

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Step 9.4.1

Simplify the numerator.

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Step 9.4.1.1

Rewrite.

$(- \frac{e^{- 3 x} \sin (5 x)}{3} - \frac{e^{- 3 x} (\cos (5 x) \cdot 5)}{9}) \frac{9}{34} + C$

Step 9.4.1.2

Move $5$ to the left of $\cos (5 x)$ .

$(- \frac{e^{- 3 x} \sin (5 x)}{3} - \frac{e^{- 3 x} (5 \cdot \cos (5 x))}{9}) \frac{9}{34} + C$

Step 9.4.1.3

Remove unnecessary parentheses.

$(- \frac{e^{- 3 x} \sin (5 x)}{3} - \frac{e^{- 3 x} \cdot 5 \cos (5 x)}{9}) \frac{9}{34} + C$

Step 9.4.2

Move $5$ to the left of $e^{- 3 x}$ .

$(- \frac{1}{3} e^{- 3 x} \sin (5 x) - \frac{5}{9} e^{- 3 x} \cos (5 x)) \frac{9}{34} + C$