Calculus Examples

$\int e^{4 x} \cos (3 x) d x$

Step 1

Reorder $e^{4 x}$ and $\cos (3 x)$ .

$\int \cos (3 x) e^{4 x} d x$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \cos (3 x)$ and $d v = e^{4 x}$ .

$\cos (3 x) (\frac{1}{4} e^{4 x}) - \int \frac{1}{4} e^{4 x} (- 3 \sin (3 x)) d x$

Step 3

Since $\frac{1}{4} \cdot - 3$ is constant with respect to $x$ , move $\frac{1}{4} \cdot - 3$ out of the integral.

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

Step 4

Combine fractions.

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Step 4.1

Combine $\frac{1}{4}$ and $e^{4 x}$ .

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

Step 4.2

Combine $\cos (3 x)$ and $\frac{e^{4 x}}{4}$ .

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

Step 4.3

Combine $\frac{1}{4}$ and $- 3$ .

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

Step 4.4

Reorder $e^{4 x}$ and $\sin (3 x)$ .

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

Step 5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \sin (3 x)$ and $d v = e^{4 x}$ .

$\frac{\cos (3 x) e^{4 x}}{4} - (\frac{- 3}{4} (\sin (3 x) (\frac{1}{4} e^{4 x}) - \int \frac{1}{4} e^{4 x} (3 \cos (3 x)) d x))$

Step 6

Since $\frac{1}{4} \cdot 3$ is constant with respect to $x$ , move $\frac{1}{4} \cdot 3$ out of the integral.

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

Step 7

Simplify terms.

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Step 7.1

Combine $\frac{1}{4}$ and $e^{4 x}$ .

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

Step 7.2

Combine $\sin (3 x)$ and $\frac{e^{4 x}}{4}$ .

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

Step 7.3

Combine $\frac{1}{4}$ and $3$ .

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

Step 7.4

Apply the distributive property.

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

Step 7.5

Multiply $\frac{\sin (3 x) e^{4 x}}{4}$ by $\frac{- 3}{4}$ .

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

Step 7.6

Multiply.

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Step 7.6.1

Multiply $4$ by $4$ .

$\frac{\cos (3 x) e^{4 x}}{4} - \frac{\sin (3 x) e^{4 x} \cdot - 3}{16} - \frac{- 3}{4} (- (\frac{3}{4} \int e^{4 x} (\cos (3 x)) d x))$

Step 7.6.2

Multiply $- 1$ by $- 1$ .

$\frac{\cos (3 x) e^{4 x}}{4} - \frac{\sin (3 x) e^{4 x} \cdot - 3}{16} + 1 (\frac{- 3}{4}) (\frac{3}{4} \int e^{4 x} (\cos (3 x)) d x)$

Step 7.6.3

Multiply $\frac{- 3}{4}$ by $1$ .

$\frac{\cos (3 x) e^{4 x}}{4} - \frac{\sin (3 x) e^{4 x} \cdot - 3}{16} + \frac{- 3}{4} (\frac{3}{4} \int e^{4 x} (\cos (3 x)) d x)$

Step 7.7

Multiply $\frac{3}{4}$ by $\frac{- 3}{4}$ .

$\frac{\cos (3 x) e^{4 x}}{4} - \frac{\sin (3 x) e^{4 x} \cdot - 3}{16} + \frac{3 \cdot - 3}{4 \cdot 4} \int e^{4 x} (\cos (3 x)) d x$

Step 7.8

Multiply.

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Step 7.8.1

Multiply $3$ by $- 3$ .

$\frac{\cos (3 x) e^{4 x}}{4} - \frac{\sin (3 x) e^{4 x} \cdot - 3}{16} + \frac{- 9}{4 \cdot 4} \int e^{4 x} (\cos (3 x)) d x$

Step 7.8.2

Multiply $4$ by $4$ .

$\frac{\cos (3 x) e^{4 x}}{4} - \frac{\sin (3 x) e^{4 x} \cdot - 3}{16} + \frac{- 9}{16} \int e^{4 x} (\cos (3 x)) d x$

Step 8

Solving for $\int e^{4 x} \cos (3 x) d x$ , we find that $\int e^{4 x} \cos (3 x) d x$ = $\frac{\frac{\cos (3 x) e^{4 x}}{4} - \frac{\sin (3 x) e^{4 x} \cdot - 3}{16}}{\frac{25}{16}}$ .

$\frac{\frac{\cos (3 x) e^{4 x}}{4} - \frac{\sin (3 x) e^{4 x} \cdot - 3}{16}}{\frac{25}{16}} + C$

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 $\sin (3 x) e^{4 x}$ .

$\frac{\frac{\cos (3 x) e^{4 x}}{4} - \frac{- 3 \cdot (\sin (3 x) e^{4 x})}{16}}{\frac{25}{16}} + C$

Step 9.1.2

Move the negative in front of the fraction.

$\frac{\frac{\cos (3 x) e^{4 x}}{4} - - \frac{3 \sin (3 x) e^{4 x}}{16}}{\frac{25}{16}} + C$

Step 9.1.3

Multiply $- 1$ by $- 1$ .

$\frac{\frac{\cos (3 x) e^{4 x}}{4} + 1 \frac{3 \sin (3 x) e^{4 x}}{16}}{\frac{25}{16}} + C$

Step 9.1.4

Multiply $\frac{3 \sin (3 x) e^{4 x}}{16}$ by $1$ .

$\frac{\frac{\cos (3 x) e^{4 x}}{4} + \frac{3 \sin (3 x) e^{4 x}}{16}}{\frac{25}{16}} + C$

Step 9.1.5

Multiply by the reciprocal of the fraction to divide by $\frac{25}{16}$ .

$(\frac{\cos (3 x) e^{4 x}}{4} + \frac{3 \sin (3 x) e^{4 x}}{16}) \frac{16}{25} + C$

Step 9.2

Rewrite $(\frac{\cos (3 x) e^{4 x}}{4} + \frac{3 \sin (3 x) e^{4 x}}{16}) \frac{16}{25} + C$ as $(\frac{1}{4} \cos (3 x) e^{4 x} + \frac{3}{16} \sin (3 x) e^{4 x}) \frac{16}{25} + C$ .

$(\frac{1}{4} \cos (3 x) e^{4 x} + \frac{3}{16} \sin (3 x) e^{4 x}) \frac{16}{25} + C$