Calculus Examples

$\int e^{2 x} \cos (4 x) d x$

Step 1

Reorder $e^{2 x}$ and $\cos (4 x)$ .

$\int \cos (4 x) e^{2 x} d x$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \cos (4 x)$ and $d v = e^{2 x}$ .

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

Step 3

Since $\frac{1}{2} \cdot - 4$ is constant with respect to $x$ , move $\frac{1}{2} \cdot - 4$ out of the integral.

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

Step 4

Combine fractions.

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Step 4.1

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

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

Step 4.2

Combine $\cos (4 x)$ and $\frac{e^{2 x}}{2}$ .

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

Step 4.3

Combine $\frac{1}{2}$ and $- 4$ .

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

Step 4.4

Reorder $e^{2 x}$ and $\sin (4 x)$ .

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

Step 5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \sin (4 x)$ and $d v = e^{2 x}$ .

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

Step 6

Since $\frac{1}{2} \cdot 4$ is constant with respect to $x$ , move $\frac{1}{2} \cdot 4$ out of the integral.

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

Step 7

Simplify terms.

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Step 7.1

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

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

Step 7.2

Combine $\sin (4 x)$ and $\frac{e^{2 x}}{2}$ .

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

Step 7.3

Combine $\frac{1}{2}$ and $4$ .

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

Step 7.4

Apply the distributive property.

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

Step 7.5

Multiply $\frac{\sin (4 x) e^{2 x}}{2}$ by $\frac{- 4}{2}$ .

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

Step 7.6

Multiply.

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Step 7.6.1

Multiply $2$ by $2$ .

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

Step 7.6.2

Multiply $- 1$ by $- 1$ .

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

Step 7.6.3

Multiply $\frac{- 4}{2}$ by $1$ .

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

Step 7.7

Multiply $\frac{4}{2}$ by $\frac{- 4}{2}$ .

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

Step 7.8

Multiply.

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Step 7.8.1

Multiply $4$ by $- 4$ .

$\frac{\cos (4 x) e^{2 x}}{2} - \frac{\sin (4 x) e^{2 x} \cdot - 4}{4} + \frac{- 16}{2 \cdot 2} \int e^{2 x} (\cos (4 x)) d x$

Step 7.8.2

Multiply $2$ by $2$ .

$\frac{\cos (4 x) e^{2 x}}{2} - \frac{\sin (4 x) e^{2 x} \cdot - 4}{4} + \frac{- 16}{4} \int e^{2 x} (\cos (4 x)) d x$

Step 8

Solving for $\int e^{2 x} \cos (4 x) d x$ , we find that $\int e^{2 x} \cos (4 x) d x$ = $\frac{\frac{\cos (4 x) e^{2 x}}{2} - \frac{\sin (4 x) e^{2 x} \cdot - 4}{4}}{\frac{20}{4}}$ .

$\frac{\frac{\cos (4 x) e^{2 x}}{2} - \frac{\sin (4 x) e^{2 x} \cdot - 4}{4}}{\frac{20}{4}} + C$

Step 9

Simplify the answer.

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Step 9.1

Simplify.

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Step 9.1.1

Move $- 4$ to the left of $\sin (4 x) e^{2 x}$ .

$\frac{\frac{\cos (4 x) e^{2 x}}{2} - \frac{- 4 \cdot (\sin (4 x) e^{2 x})}{4}}{\frac{20}{4}} + C$

Step 9.1.2

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

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Step 9.1.2.1

Factor $4$ out of $- 4 \sin (4 x) e^{2 x}$ .

$\frac{\frac{\cos (4 x) e^{2 x}}{2} - \frac{4 (- \sin (4 x) e^{2 x})}{4}}{\frac{20}{4}} + C$

Step 9.1.2.2

Cancel the common factors.

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Step 9.1.2.2.1

Factor $4$ out of $4$ .

$\frac{\frac{\cos (4 x) e^{2 x}}{2} - \frac{4 (- \sin (4 x) e^{2 x})}{4 (1)}}{\frac{20}{4}} + C$

Step 9.1.2.2.2

Cancel the common factor.

$\frac{\frac{\cos (4 x) e^{2 x}}{2} - \frac{4 (- \sin (4 x) e^{2 x})}{4 \cdot 1}}{\frac{20}{4}} + C$

Step 9.1.2.2.3

Rewrite the expression.

$\frac{\frac{\cos (4 x) e^{2 x}}{2} - \frac{- \sin (4 x) e^{2 x}}{1}}{\frac{20}{4}} + C$

Step 9.1.2.2.4

Divide $- \sin (4 x) e^{2 x}$ by $1$ .

$\frac{\frac{\cos (4 x) e^{2 x}}{2} - (- \sin (4 x) e^{2 x})}{\frac{20}{4}} + C$

Step 9.1.3

Multiply $- 1$ by $- 1$ .

$\frac{\frac{\cos (4 x) e^{2 x}}{2} + 1 (\sin (4 x) e^{2 x})}{\frac{20}{4}} + C$

Step 9.1.4

Multiply $\sin (4 x)$ by $1$ .

$\frac{\frac{\cos (4 x) e^{2 x}}{2} + \sin (4 x) e^{2 x}}{\frac{20}{4}} + C$

Step 9.1.5

Cancel the common factor of $20$ and $4$ .

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Step 9.1.5.1

Factor $4$ out of $20$ .

$\frac{\frac{\cos (4 x) e^{2 x}}{2} + \sin (4 x) e^{2 x}}{\frac{4 \cdot 5}{4}} + C$

Step 9.1.5.2

Cancel the common factors.

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Step 9.1.5.2.1

Factor $4$ out of $4$ .

$\frac{\frac{\cos (4 x) e^{2 x}}{2} + \sin (4 x) e^{2 x}}{\frac{4 \cdot 5}{4 (1)}} + C$

Step 9.1.5.2.2

Cancel the common factor.

$\frac{\frac{\cos (4 x) e^{2 x}}{2} + \sin (4 x) e^{2 x}}{\frac{4 \cdot 5}{4 \cdot 1}} + C$

Step 9.1.5.2.3

Rewrite the expression.

$\frac{\frac{\cos (4 x) e^{2 x}}{2} + \sin (4 x) e^{2 x}}{\frac{5}{1}} + C$

Step 9.1.5.2.4

Divide $5$ by $1$ .

$\frac{\frac{\cos (4 x) e^{2 x}}{2} + \sin (4 x) e^{2 x}}{5} + C$

Step 9.1.6

Multiply $\frac{\frac{\cos (4 x) e^{2 x}}{2} + \sin (4 x) e^{2 x}}{5}$ by $\frac{2}{2}$ .

$\frac{2}{2} \cdot \frac{\frac{\cos (4 x) e^{2 x}}{2} + \sin (4 x) e^{2 x}}{5} + C$

Step 9.1.7

Combine.

$\frac{2 (\frac{\cos (4 x) e^{2 x}}{2} + \sin (4 x) e^{2 x})}{2 \cdot 5} + C$

Step 9.1.8

Apply the distributive property.

$\frac{2 \frac{\cos (4 x) e^{2 x}}{2} + 2 (\sin (4 x) e^{2 x})}{2 \cdot 5} + C$

Step 9.1.9

Cancel the common factor of $2$ .

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Step 9.1.9.1

Cancel the common factor.

$\frac{2 \frac{\cos (4 x) e^{2 x}}{2} + 2 (\sin (4 x) e^{2 x})}{2 \cdot 5} + C$

Step 9.1.9.2

Rewrite the expression.

$\frac{\cos (4 x) e^{2 x} + 2 (\sin (4 x) e^{2 x})}{2 \cdot 5} + C$

Step 9.1.10

Multiply $2$ by $5$ .

$\frac{\cos (4 x) e^{2 x} + 2 \sin (4 x) e^{2 x}}{10} + C$

Step 9.2

Rewrite $\frac{\cos (4 x) e^{2 x} + 2 \sin (4 x) e^{2 x}}{10} + C$ as $\frac{1}{10} (\cos (4 x) e^{2 x} + 2 \sin (4 x) e^{2 x}) + C$ .

$\frac{1}{10} (\cos (4 x) e^{2 x} + 2 \sin (4 x) e^{2 x}) + C$