Calculus Examples

$\int_{0}^{\frac{π}{2}} e^{- \cos (x)} \sin (x) d x$

Step 1

Let $u = - \cos (x)$ . Then $d u = \sin (x) d x$ , so $\frac{1}{\sin (x)} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = - \cos (x)$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $- \cos (x)$ .

$\frac{d}{d x} [- \cos (x)]$

Step 1.1.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- \cos (x)$ with respect to $x$ is $- \frac{d}{d x} [\cos (x)]$ .

$- \frac{d}{d x} [\cos (x)]$

Step 1.1.3

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$- - \sin (x)$

Step 1.1.4

Multiply.

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Step 1.1.4.1

Multiply $- 1$ by $- 1$ .

$1 \sin (x)$

Step 1.1.4.2

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

$\sin (x)$

Step 1.2

Substitute the lower limit in for $x$ in $u = - \cos (x)$ .

$u_{lower} = - \cos (0)$

Step 1.3

Simplify.

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Step 1.3.1

The exact value of $\cos (0)$ is $1$ .

$u_{lower} = - 1 \cdot 1$

Step 1.3.2

Multiply $- 1$ by $1$ .

$u_{lower} = - 1$

Step 1.4

Substitute the upper limit in for $x$ in $u = - \cos (x)$ .

$u_{upper} = - \cos (\frac{π}{2})$

Step 1.5

Simplify.

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Step 1.5.1

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$u_{upper} = - 0$

Step 1.5.2

Multiply $- 1$ by $0$ .

$u_{upper} = 0$

Step 1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = - 1$

$u_{upper} = 0$

Step 1.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{- 1}^{0} e^{u} d u$

Step 2

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$e^{u}]_{- 1}^{0}$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $e^{u}$ at $0$ and at $- 1$ .

$(e^{0}) - e^{- 1}$

Step 3.2

Anything raised to $0$ is $1$ .

$1 - e^{- 1}$

Step 4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$1 - \frac{1}{e}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$1 - \frac{1}{e}$

Decimal Form:

$0.63212055 \dots$