Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify the expression.

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Step 4.1

Multiply $- 1$ by $4$ .

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

Step 4.2

Multiply $- 4$ by $- 1$ .

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

Step 4.3

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

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

Step 5

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

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

Step 6

Since $- - 4$ is constant with respect to $x$ , move $- - 4$ out of the integral.

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

Step 7

Simplify by multiplying through.

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Step 7.1

Multiply $- 1$ by $- 4$ .

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

Step 7.2

Multiply $4$ by $- 1$ .

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

Step 7.3

Apply the distributive property.

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

Step 7.4

Multiply.

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Step 7.4.1

Multiply $- 1$ by $4$ .

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

Step 7.4.2

Multiply $- 4$ by $4$ .

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

Step 8

Solving for $\int e^{- x} \sin (4 x) d x$ , we find that $\int e^{- x} \sin (4 x) d x$ = $\frac{\sin (4 x) (- e^{- x}) - 4 (\cos (4 x) (e^{- x}))}{17}$ .

$\frac{\sin (4 x) (- e^{- x}) - 4 (\cos (4 x) (e^{- x}))}{17} + C$

Step 9

Simplify the answer.

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Step 9.1

Rewrite $\frac{\sin (4 x) (- e^{- x}) - 4 \cos (4 x) e^{- x}}{17} + C$ as $\frac{- \sin (4 x) e^{- x} - 4 \cos (4 x) e^{- x}}{17} + C$ .

$\frac{- \sin (4 x) e^{- x} - 4 \cos (4 x) e^{- x}}{17} + C$

Step 9.2

Simplify.

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Step 9.2.1

Simplify the numerator.

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Step 9.2.1.1

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

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Step 9.2.1.1.1

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

$\frac{e^{- x} (- \sin (4 x)) - 4 \cos (4 x) e^{- x}}{17} + C$

Step 9.2.1.1.2

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

$\frac{e^{- x} (- \sin (4 x)) + e^{- x} (- 4 \cos (4 x))}{17} + C$

Step 9.2.1.1.3

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

$\frac{e^{- x} (- \sin (4 x) - 4 \cos (4 x))}{17} + C$

Step 9.2.1.2

Factor $- 1$ out of $- \sin (4 x) - 4 \cos (4 x)$ .

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Step 9.2.1.2.1

Factor $- 1$ out of $- \sin (4 x)$ .

$\frac{e^{- x} (- (\sin (4 x)) - 4 \cos (4 x))}{17} + C$

Step 9.2.1.2.2

Factor $- 1$ out of $- 4 \cos (4 x)$ .

$\frac{e^{- x} (- (\sin (4 x)) - (4 \cos (4 x)))}{17} + C$

Step 9.2.1.2.3

Factor $- 1$ out of $- (\sin (4 x)) - (4 \cos (4 x))$ .

$\frac{e^{- x} (- (\sin (4 x) + 4 \cos (4 x)))}{17} + C$

$\frac{e^{- x} \cdot - 1 (\sin (4 x) + 4 \cos (4 x))}{17} + C$

Step 9.2.1.3

Factor out negative.

$\frac{- e^{- x} (\sin (4 x) + 4 \cos (4 x))}{17} + C$

Step 9.2.2

Move the negative in front of the fraction.

$- \frac{1}{17} e^{- x} (\sin (4 x) + 4 \cos (4 x)) + C$