Calculus Examples

\int e^{x} \sin (x) d x

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

Step 1

Reorder

e^{x}

$e^{x}$ and

\sin (x)

$\sin (x)$ .

\int \sin (x) e^{x} d x

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

Step 2

Integrate by parts using the formula

\int u d v = u v - \int v d u

$\int u d v = u v - \int v d u$ , where

u = \sin (x)

$u = \sin (x)$ and

d v = e^{x}

$d v = e^{x}$ .

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

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

Step 3

Reorder

e^{x}

$e^{x}$ and

\cos (x)

$\cos (x)$ .

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

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

Step 4

Integrate by parts using the formula

\int u d v = u v - \int v d u

$\int u d v = u v - \int v d u$ , where

u = \cos (x)

$u = \cos (x)$ and

d v = e^{x}

$d v = e^{x}$ .

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

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

Step 5

Since

- 1

$- 1$ is constant with respect to

x

$x$ , move

- 1

$- 1$ out of the integral.

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

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

Step 6

Simplify by multiplying through.

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Step 6.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

\sin (x) e^{x} - (\cos (x) e^{x} + 1 \int e^{x} (\sin (x)) d x)

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

Step 6.2

Multiply

\int e^{x} (\sin (x)) d x

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

1

$1$ .

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

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

Step 6.3

Apply the distributive property.

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

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

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

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

Step 7

Solving for

\int e^{x} \sin (x) d x

$\int e^{x} \sin (x) d x$ , we find that

\int e^{x} \sin (x) d x

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

\frac{\sin (x) e^{x} - (\cos (x) e^{x})}{2}

$\frac{\sin (x) e^{x} - (\cos (x) e^{x})}{2}$ .

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

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

Step 8

Rewrite

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

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

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

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

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

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