Calculus Examples

$\int e^{2 θ} \sin (3 θ) d θ$

Step 1

Reorder $e^{2 θ}$ and $\sin (3 θ)$ .

$\int \sin (3 θ) e^{2 θ} d θ$

Step 2

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

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

Step 3

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

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

Step 4

Combine fractions.

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Step 4.1

Combine $\frac{1}{2}$ and $e^{2 θ}$ .

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Reorder $e^{2 θ}$ and $\cos (3 θ)$ .

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify terms.

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Step 7.1

Combine $\frac{1}{2}$ and $e^{2 θ}$ .

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

Apply the distributive property.

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

Step 7.5

Multiply $\frac{\cos (3 θ) e^{2 θ}}{2}$ by $\frac{3}{2}$ .

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

Step 7.6

Multiply.

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Step 7.6.1

Multiply $2$ by $2$ .

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

Step 7.6.2

Multiply $- 1$ by $- 1$ .

$\frac{\sin (3 θ) e^{2 θ}}{2} - \frac{\cos (3 θ) e^{2 θ} \cdot 3}{4} + 1 (\frac{3}{2}) (\frac{- 3}{2} \int e^{2 θ} (\sin (3 θ)) d θ)$

Step 7.6.3

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

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

Step 7.7

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

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

Step 7.8

Multiply.

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Step 7.8.1

Multiply $- 3$ by $3$ .

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

Step 7.8.2

Multiply $2$ by $2$ .

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

Step 8

Solving for $\int e^{2 θ} \sin (3 θ) d θ$ , we find that $\int e^{2 θ} \sin (3 θ) d θ$ = $\frac{\frac{\sin (3 θ) e^{2 θ}}{2} - \frac{\cos (3 θ) e^{2 θ} \cdot 3}{4}}{\frac{13}{4}}$ .

$\frac{\frac{\sin (3 θ) e^{2 θ}}{2} - \frac{\cos (3 θ) e^{2 θ} \cdot 3}{4}}{\frac{13}{4}} + C$

Step 9

Simplify the answer.

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Step 9.1

Simplify.

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Step 9.1.1

Move $3$ to the left of $\cos (3 θ) e^{2 θ}$ .

$\frac{\frac{\sin (3 θ) e^{2 θ}}{2} - \frac{3 \cdot (\cos (3 θ) e^{2 θ})}{4}}{\frac{13}{4}} + C$

Step 9.1.2

Multiply by the reciprocal of the fraction to divide by $\frac{13}{4}$ .

$(\frac{\sin (3 θ) e^{2 θ}}{2} - \frac{3 \cos (3 θ) e^{2 θ}}{4}) \frac{4}{13} + C$

Step 9.2

Rewrite $(\frac{\sin (3 θ) e^{2 θ}}{2} - \frac{3 \cos (3 θ) e^{2 θ}}{4}) \frac{4}{13} + C$ as $(\frac{1}{2} \sin (3 θ) e^{2 θ} - \frac{3}{4} \cos (3 θ) e^{2 θ}) \frac{4}{13} + C$ .

$(\frac{1}{2} \sin (3 θ) e^{2 θ} - \frac{3}{4} \cos (3 θ) e^{2 θ}) \frac{4}{13} + C$

Step 9.3

Simplify.

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Step 9.3.1

Combine $\frac{1}{2}$ and $\sin (3 θ)$ .

$(\frac{\sin (3 θ)}{2} e^{2 θ} - \frac{3}{4} \cos (3 θ) e^{2 θ}) \frac{4}{13} + C$

Step 9.3.2

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

$(\frac{\sin (3 θ) e^{2 θ}}{2} - \frac{3}{4} \cos (3 θ) e^{2 θ}) \frac{4}{13} + C$

Step 9.3.3

Combine $\cos (3 θ)$ and $\frac{3}{4}$ .

$(\frac{\sin (3 θ) e^{2 θ}}{2} - \frac{\cos (3 θ) \cdot 3}{4} e^{2 θ}) \frac{4}{13} + C$

Step 9.3.4

Combine $e^{2 θ}$ and $\frac{\cos (3 θ) \cdot 3}{4}$ .

$(\frac{\sin (3 θ) e^{2 θ}}{2} - \frac{e^{2 θ} (\cos (3 θ) \cdot 3)}{4}) \frac{4}{13} + C$

Step 9.3.5

Move $3$ to the left of $\cos (3 θ)$ .

$(\frac{\sin (3 θ) e^{2 θ}}{2} - \frac{e^{2 θ} (3 \cos (3 θ))}{4}) \frac{4}{13} + C$

Step 9.4

Simplify.

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Step 9.4.1

Reorder factors in $\frac{\sin (3 θ) e^{2 θ}}{2}$ .

$(\frac{e^{2 θ} \sin (3 θ)}{2} - \frac{e^{2 θ} (3 \cos (3 θ))}{4}) \frac{4}{13} + C$

Step 9.4.2

Simplify the numerator.

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Step 9.4.2.1

Rewrite.

$(\frac{e^{2 θ} \sin (3 θ)}{2} - \frac{e^{2 θ} (\cos (3 θ) \cdot 3)}{4}) \frac{4}{13} + C$

Step 9.4.2.2

Move $3$ to the left of $\cos (3 θ)$ .

$(\frac{e^{2 θ} \sin (3 θ)}{2} - \frac{e^{2 θ} (3 \cdot \cos (3 θ))}{4}) \frac{4}{13} + C$

Step 9.4.2.3

Remove unnecessary parentheses.

$(\frac{e^{2 θ} \sin (3 θ)}{2} - \frac{e^{2 θ} \cdot 3 \cos (3 θ)}{4}) \frac{4}{13} + C$

Step 9.4.3

Move $3$ to the left of $e^{2 θ}$ .

$(\frac{1}{2} e^{2 θ} \sin (3 θ) - \frac{3}{4} e^{2 θ} \cos (3 θ)) \frac{4}{13} + C$