Calculus Examples

$\int e^{2 θ} \sin (2 θ) d θ$

Step 1

Reorder $e^{2 θ}$ and $\sin (2 θ)$ .

$\int \sin (2 θ) e^{2 θ} d θ$

Step 2

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

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

Step 3

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

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

Step 4

Combine fractions.

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Step 4.1

Combine $\frac{1}{2}$ and $e^{2 θ}$ .

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Reorder $e^{2 θ}$ and $\cos (2 θ)$ .

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify terms.

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Step 7.1

Combine $\frac{1}{2}$ and $e^{2 θ}$ .

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

Apply the distributive property.

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

Step 7.5

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

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

Step 7.6

Multiply.

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Step 7.6.1

Multiply $2$ by $2$ .

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

Step 7.6.2

Multiply $- 1$ by $- 1$ .

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

Step 7.6.3

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

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

Step 7.7

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

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

Step 7.8

Multiply.

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Step 7.8.1

Multiply $- 2$ by $2$ .

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

Step 7.8.2

Multiply $2$ by $2$ .

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

Step 8

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

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

Step 9

Simplify the answer.

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Step 9.1

Simplify.

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Step 9.1.1

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

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

Step 9.1.2

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

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Step 9.1.2.1

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

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

Step 9.1.2.2

Cancel the common factors.

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Step 9.1.2.2.1

Factor $2$ out of $4$ .

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

Step 9.1.2.2.2

Cancel the common factor.

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

Step 9.1.2.2.3

Rewrite the expression.

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

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 $8$ .

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

Step 9.1.3.2

Cancel the common factors.

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Step 9.1.3.2.1

Factor $4$ out of $4$ .

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

Step 9.1.3.2.2

Cancel the common factor.

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

Step 9.1.3.2.3

Rewrite the expression.

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

Step 9.1.3.2.4

Divide $2$ by $1$ .

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

Step 9.1.4

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

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

Step 9.1.5

Combine.

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

Step 9.1.6

Apply the distributive property.

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

Step 9.1.7

Cancel the common factor of $2$ .

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Step 9.1.7.1

Cancel the common factor.

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

Step 9.1.7.2

Rewrite the expression.

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

Step 9.1.8

Multiply $- 1$ by $2$ .

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

Step 9.1.9

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

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

Step 9.1.10

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

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Step 9.1.10.1

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

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

Step 9.1.10.2

Cancel the common factors.

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Step 9.1.10.2.1

Factor $2$ out of $2$ .

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

Step 9.1.10.2.2

Cancel the common factor.

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

Step 9.1.10.2.3

Rewrite the expression.

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

Step 9.1.10.2.4

Divide $- \cos (2 θ) e^{2 θ}$ by $1$ .

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

Step 9.1.11

Multiply $2$ by $2$ .

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

Step 9.2

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

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