Calculus Examples

$e^{x} (- \sin (x))$

Step 1

Write $e^{x} (- \sin (x))$ as a function.

$f (x) = e^{x} (- \sin (x))$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Simplify by multiplying through.

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Step 10.1

Multiply $- 1$ by $- 1$ .

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

Step 10.2

Multiply $\int e^{x} (\sin (x)) d x$ by $1$ .

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

Step 10.3

Apply the distributive property.

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

Step 11

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

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

Step 12

Simplify the answer.

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Step 12.1

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

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

Step 12.2

Remove parentheses.

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

Step 13

The answer is the antiderivative of the function $f (x) = e^{x} (- \sin (x))$ .

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