Calculus Examples

$\int e^{8 θ} \sin (9 θ) d θ$

Step 1

Reorder $e^{8 θ}$ and $\sin (9 θ)$ .

$\int \sin (9 θ) e^{8 θ} d θ$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \sin (9 θ)$ and $d v = e^{8 θ}$ .

$\sin (9 θ) (\frac{1}{8} e^{8 θ}) - \int \frac{1}{8} e^{8 θ} (9 \cos (9 θ)) d θ$

Step 3

Since $\frac{1}{8} \cdot 9$ is constant with respect to $θ$ , move $\frac{1}{8} \cdot 9$ out of the integral.

$\sin (9 θ) (\frac{1}{8} e^{8 θ}) - (\frac{1}{8} \cdot 9 \int e^{8 θ} (\cos (9 θ)) d θ)$

Step 4

Combine fractions.

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Step 4.1

Combine $\frac{1}{8}$ and $e^{8 θ}$ .

$\sin (9 θ) \frac{e^{8 θ}}{8} - (\frac{1}{8} \cdot 9 \int e^{8 θ} (\cos (9 θ)) d θ)$

Step 4.2

Combine $\sin (9 θ)$ and $\frac{e^{8 θ}}{8}$ .

$\frac{\sin (9 θ) e^{8 θ}}{8} - (\frac{1}{8} \cdot 9 \int e^{8 θ} (\cos (9 θ)) d θ)$

Step 4.3

Combine $\frac{1}{8}$ and $9$ .

$\frac{\sin (9 θ) e^{8 θ}}{8} - (\frac{9}{8} \int e^{8 θ} (\cos (9 θ)) d θ)$

Step 4.4

Reorder $e^{8 θ}$ and $\cos (9 θ)$ .

$\frac{\sin (9 θ) e^{8 θ}}{8} - (\frac{9}{8} \int \cos (9 θ) e^{8 θ} d θ)$

Step 5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \cos (9 θ)$ and $d v = e^{8 θ}$ .

$\frac{\sin (9 θ) e^{8 θ}}{8} - (\frac{9}{8} (\cos (9 θ) (\frac{1}{8} e^{8 θ}) - \int \frac{1}{8} e^{8 θ} (- 9 \sin (9 θ)) d θ))$

Step 6

Since $\frac{1}{8} \cdot - 9$ is constant with respect to $θ$ , move $\frac{1}{8} \cdot - 9$ out of the integral.

$\frac{\sin (9 θ) e^{8 θ}}{8} - (\frac{9}{8} (\cos (9 θ) (\frac{1}{8} e^{8 θ}) - (\frac{1}{8} \cdot - 9 \int e^{8 θ} (\sin (9 θ)) d θ)))$

Step 7

Simplify terms.

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Step 7.1

Combine $\frac{1}{8}$ and $e^{8 θ}$ .

$\frac{\sin (9 θ) e^{8 θ}}{8} - (\frac{9}{8} (\cos (9 θ) \frac{e^{8 θ}}{8} - (\frac{1}{8} \cdot - 9 \int e^{8 θ} (\sin (9 θ)) d θ)))$

Step 7.2

Combine $\cos (9 θ)$ and $\frac{e^{8 θ}}{8}$ .

$\frac{\sin (9 θ) e^{8 θ}}{8} - (\frac{9}{8} (\frac{\cos (9 θ) e^{8 θ}}{8} - (\frac{1}{8} \cdot - 9 \int e^{8 θ} (\sin (9 θ)) d θ)))$

Step 7.3

Combine $\frac{1}{8}$ and $- 9$ .

$\frac{\sin (9 θ) e^{8 θ}}{8} - (\frac{9}{8} (\frac{\cos (9 θ) e^{8 θ}}{8} - (\frac{- 9}{8} \int e^{8 θ} (\sin (9 θ)) d θ)))$

Step 7.4

Apply the distributive property.

$\frac{\sin (9 θ) e^{8 θ}}{8} - \frac{9}{8} \cdot \frac{\cos (9 θ) e^{8 θ}}{8} - \frac{9}{8} (- (\frac{- 9}{8} \int e^{8 θ} (\sin (9 θ)) d θ))$

Step 7.5

Multiply $\frac{\cos (9 θ) e^{8 θ}}{8}$ by $\frac{9}{8}$ .

$\frac{\sin (9 θ) e^{8 θ}}{8} - \frac{\cos (9 θ) e^{8 θ} \cdot 9}{8 \cdot 8} - \frac{9}{8} (- (\frac{- 9}{8} \int e^{8 θ} (\sin (9 θ)) d θ))$

Step 7.6

Multiply.

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Step 7.6.1

Multiply $8$ by $8$ .

$\frac{\sin (9 θ) e^{8 θ}}{8} - \frac{\cos (9 θ) e^{8 θ} \cdot 9}{64} - \frac{9}{8} (- (\frac{- 9}{8} \int e^{8 θ} (\sin (9 θ)) d θ))$

Step 7.6.2

Multiply $- 1$ by $- 1$ .

$\frac{\sin (9 θ) e^{8 θ}}{8} - \frac{\cos (9 θ) e^{8 θ} \cdot 9}{64} + 1 (\frac{9}{8}) (\frac{- 9}{8} \int e^{8 θ} (\sin (9 θ)) d θ)$

Step 7.6.3

Multiply $\frac{9}{8}$ by $1$ .

$\frac{\sin (9 θ) e^{8 θ}}{8} - \frac{\cos (9 θ) e^{8 θ} \cdot 9}{64} + \frac{9}{8} (\frac{- 9}{8} \int e^{8 θ} (\sin (9 θ)) d θ)$

Step 7.7

Multiply $\frac{- 9}{8}$ by $\frac{9}{8}$ .

$\frac{\sin (9 θ) e^{8 θ}}{8} - \frac{\cos (9 θ) e^{8 θ} \cdot 9}{64} + \frac{- 9 \cdot 9}{8 \cdot 8} \int e^{8 θ} (\sin (9 θ)) d θ$

Step 7.8

Multiply.

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Step 7.8.1

Multiply $- 9$ by $9$ .

$\frac{\sin (9 θ) e^{8 θ}}{8} - \frac{\cos (9 θ) e^{8 θ} \cdot 9}{64} + \frac{- 81}{8 \cdot 8} \int e^{8 θ} (\sin (9 θ)) d θ$

Step 7.8.2

Multiply $8$ by $8$ .

$\frac{\sin (9 θ) e^{8 θ}}{8} - \frac{\cos (9 θ) e^{8 θ} \cdot 9}{64} + \frac{- 81}{64} \int e^{8 θ} (\sin (9 θ)) d θ$

Step 8

Solving for $\int e^{8 θ} \sin (9 θ) d θ$ , we find that $\int e^{8 θ} \sin (9 θ) d θ$ = $\frac{\frac{\sin (9 θ) e^{8 θ}}{8} - \frac{\cos (9 θ) e^{8 θ} \cdot 9}{64}}{\frac{145}{64}}$ .

$\frac{\frac{\sin (9 θ) e^{8 θ}}{8} - \frac{\cos (9 θ) e^{8 θ} \cdot 9}{64}}{\frac{145}{64}} + C$

Step 9

Simplify the answer.

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Step 9.1

Simplify the answer.

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Step 9.1.1

Simplify.

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Step 9.1.1.1

Move $9$ to the left of $\cos (9 θ) e^{8 θ}$ .

$\frac{\frac{\sin (9 θ) e^{8 θ}}{8} - \frac{9 \cdot (\cos (9 θ) e^{8 θ})}{64}}{\frac{145}{64}} + C$

Step 9.1.1.2

Multiply by the reciprocal of the fraction to divide by $\frac{145}{64}$ .

$(\frac{\sin (9 θ) e^{8 θ}}{8} - \frac{9 \cos (9 θ) e^{8 θ}}{64}) \frac{64}{145} + C$

Step 9.1.2

Rewrite $(\frac{\sin (9 θ) e^{8 θ}}{8} - \frac{9 \cos (9 θ) e^{8 θ}}{64}) \frac{64}{145} + C$ as $(\frac{1}{8} \sin (9 θ) e^{8 θ} - \frac{9}{64} \cos (9 θ) e^{8 θ}) \frac{64}{145} + C$ .

$(\frac{1}{8} \sin (9 θ) e^{8 θ} - \frac{9}{64} \cos (9 θ) e^{8 θ}) \frac{64}{145} + C$

Step 9.1.3

Simplify.

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Step 9.1.3.1

Combine $\frac{1}{8}$ and $\sin (9 θ)$ .

$(\frac{\sin (9 θ)}{8} e^{8 θ} - \frac{9}{64} \cos (9 θ) e^{8 θ}) \frac{64}{145} + C$

Step 9.1.3.2

Combine $\frac{\sin (9 θ)}{8}$ and $e^{8 θ}$ .

$(\frac{\sin (9 θ) e^{8 θ}}{8} - \frac{9}{64} \cos (9 θ) e^{8 θ}) \frac{64}{145} + C$

Step 9.1.3.3

Combine $\cos (9 θ)$ and $\frac{9}{64}$ .

$(\frac{\sin (9 θ) e^{8 θ}}{8} - \frac{\cos (9 θ) \cdot 9}{64} e^{8 θ}) \frac{64}{145} + C$

Step 9.1.3.4

Combine $e^{8 θ}$ and $\frac{\cos (9 θ) \cdot 9}{64}$ .

$(\frac{\sin (9 θ) e^{8 θ}}{8} - \frac{e^{8 θ} (\cos (9 θ) \cdot 9)}{64}) \frac{64}{145} + C$

Step 9.1.3.5

Move $9$ to the left of $\cos (9 θ)$ .

$(\frac{\sin (9 θ) e^{8 θ}}{8} - \frac{e^{8 θ} (9 \cos (9 θ))}{64}) \frac{64}{145} + C$

Step 9.1.4

Simplify.

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Step 9.1.4.1

Reorder factors in $\frac{\sin (9 θ) e^{8 θ}}{8}$ .

$(\frac{e^{8 θ} \sin (9 θ)}{8} - \frac{e^{8 θ} (9 \cos (9 θ))}{64}) \frac{64}{145} + C$

Step 9.1.4.2

Simplify the numerator.

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Step 9.1.4.2.1

Rewrite.

$(\frac{e^{8 θ} \sin (9 θ)}{8} - \frac{e^{8 θ} (\cos (9 θ) \cdot 9)}{64}) \frac{64}{145} + C$

Step 9.1.4.2.2

Move $9$ to the left of $\cos (9 θ)$ .

$(\frac{e^{8 θ} \sin (9 θ)}{8} - \frac{e^{8 θ} (9 \cdot \cos (9 θ))}{64}) \frac{64}{145} + C$

Step 9.1.4.2.3

Remove unnecessary parentheses.

$(\frac{e^{8 θ} \sin (9 θ)}{8} - \frac{e^{8 θ} \cdot 9 \cos (9 θ)}{64}) \frac{64}{145} + C$

Step 9.1.4.3

Move $9$ to the left of $e^{8 θ}$ .

$(\frac{1}{8} e^{8 θ} \sin (9 θ) - \frac{9}{64} e^{8 θ} \cos (9 θ)) \frac{64}{145} + C$

Step 9.2

Rewrite $(\frac{1}{8} e^{8 θ} \sin (9 θ) - \frac{9}{64} e^{8 θ} \cos (9 θ)) \frac{64}{145} + C$ as $(\frac{e^{8 θ} \sin (9 θ)}{8} - \frac{9 \cos (9 θ) e^{8 θ}}{64}) \frac{64}{145} + C$ .

$(\frac{e^{8 θ} \sin (9 θ)}{8} - \frac{9 \cos (9 θ) e^{8 θ}}{64}) \frac{64}{145} + C$

Step 9.3

Reorder factors in $\frac{9 \cos (9 θ) e^{8 θ}}{64}$ .

$(\frac{1}{8} e^{8 θ} \sin (9 θ) - \frac{9}{64} e^{8 θ} \cos (9 θ)) \frac{64}{145} + C$