Calculus Examples

$\int_{- 2}^{2} 36^{\frac{x}{2}} d x$

Step 1

Let $u = \frac{x}{2}$ . Then $d u = \frac{1}{2} d x$ , so $2 d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = \frac{x}{2}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $\frac{x}{2}$ .

$\frac{d}{d x} [\frac{x}{2}]$

Step 1.1.2

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

$\frac{1}{2} \frac{d}{d x} [x]$

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{1}{2} \cdot 1$

Step 1.1.4

Multiply $\frac{1}{2}$ by $1$ .

$\frac{1}{2}$

Step 1.2

Substitute the lower limit in for $x$ in $u = \frac{x}{2}$ .

$u_{lower} = \frac{- 2}{2}$

Step 1.3

Divide $- 2$ by $2$ .

$u_{lower} = - 1$

Step 1.4

Substitute the upper limit in for $x$ in $u = \frac{x}{2}$ .

$u_{upper} = \frac{2}{2}$

Step 1.5

Divide $2$ by $2$ .

$u_{upper} = 1$

Step 1.6

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

$u_{lower} = - 1$

$u_{upper} = 1$

Step 1.7

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

$\int_{- 1}^{1} 36^{u} \cdot \frac{1}{\frac{1}{2}} d u$

Step 2

Simplify.

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Step 2.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{2}$ .

$\int_{- 1}^{1} 36^{u} \cdot (1 \cdot 2) d u$

Step 2.2

Multiply $2$ by $1$ .

$\int_{- 1}^{1} 36^{u} \cdot 2 d u$

Step 2.3

Move $2$ to the left of $36^{u}$ .

$\int_{- 1}^{1} 2 \cdot 36^{u} d u$

Step 3

Since $2$ is constant with respect to $u$ , move $2$ out of the integral.

$2 \int_{- 1}^{1} 36^{u} d u$

Step 4

The integral of $36^{u}$ with respect to $u$ is $\frac{36^{u}}{\ln (36)}$ .

$2 (\frac{36^{u}}{\ln (36)}]_{- 1}^{1})$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $\frac{36^{u}}{\ln (36)}$ at $1$ and at $- 1$ .

$2 ((\frac{36^{1}}{\ln (36)}) - \frac{36^{- 1}}{\ln (36)})$

Step 5.2

Simplify.

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Step 5.2.1

Evaluate the exponent.

$2 (\frac{36}{\ln (36)} - \frac{36^{- 1}}{\ln (36)})$

Step 5.2.2

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

$2 (\frac{36}{\ln (36)} - \frac{\frac{1}{36}}{\ln (36)})$

Step 5.2.3

Rewrite $\frac{\frac{1}{36}}{\ln (36)}$ as a product.

$2 (\frac{36}{\ln (36)} - (\frac{1}{36} \cdot \frac{1}{\ln (36)}))$

Step 5.2.4

Multiply $\frac{1}{36}$ by $\frac{1}{\ln (36)}$ .

$2 (\frac{36}{\ln (36)} - \frac{1}{36 \ln (36)})$

Step 6

The result can be shown in multiple forms.

Exact Form:

$2 (\frac{36}{\ln (36)} - \frac{1}{36 \ln (36)})$

Decimal Form:

$20.07647948 \dots$

Step 7