Calculus Examples

$\int_{0}^{\ln (16)} e^{\frac{x}{4}} d x$

Step 1

Let $u = \frac{x}{4}$ . Then $d u = \frac{1}{4} d x$ , so $4 d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = \frac{x}{4}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $\frac{x}{4}$ .

$\frac{d}{d x} [\frac{x}{4}]$

Step 1.1.2

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{x}{4}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [x]$ .

$\frac{1}{4} \frac{d}{d x} [x]$

Step 1.1.3

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

$\frac{1}{4} \cdot 1$

Step 1.1.4

Multiply $\frac{1}{4}$ by $1$ .

$\frac{1}{4}$

Step 1.2

Substitute the lower limit in for $x$ in $u = \frac{x}{4}$ .

$u_{lower} = \frac{0}{4}$

Step 1.3

Divide $0$ by $4$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $x$ in $u = \frac{x}{4}$ .

$u_{upper} = \frac{\ln (16)}{4}$

Step 1.5

Simplify.

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Step 1.5.1

Rewrite $\ln (16)$ as $\ln (2^{4})$ .

$u_{upper} = \frac{\ln (2^{4})}{4}$

Step 1.5.2

Expand $\ln (2^{4})$ by moving $4$ outside the logarithm.

$u_{upper} = \frac{4 \ln (2)}{4}$

Step 1.5.3

Cancel the common factor of $4$ .

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Step 1.5.3.1

Cancel the common factor.

$u_{upper} = \frac{4 \ln (2)}{4}$

Step 1.5.3.2

Divide $\ln (2)$ by $1$ .

$u_{upper} = \ln (2)$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = \ln (2)$

Step 1.7

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

$\int_{0}^{\ln (2)} e^{u} \frac{1}{\frac{1}{4}} d u$

Step 2

Simplify.

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Step 2.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{4}$ .

$\int_{0}^{\ln (2)} e^{u} (1 \cdot 4) d u$

Step 2.2

Multiply $4$ by $1$ .

$\int_{0}^{\ln (2)} e^{u} \cdot 4 d u$

Step 2.3

Move $4$ to the left of $e^{u}$ .

$\int_{0}^{\ln (2)} 4 e^{u} d u$

Step 3

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

$4 \int_{0}^{\ln (2)} e^{u} d u$

Step 4

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

$4 (e^{u}]_{0}^{\ln (2)})$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $e^{u}$ at $\ln (2)$ and at $0$ .

$4 ((e^{\ln (2)}) - e^{0})$

Step 5.2

Simplify.

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Step 5.2.1

Exponentiation and log are inverse functions.

$4 (2 - e^{0})$

Step 5.2.2

Anything raised to $0$ is $1$ .

$4 (2 - 1 \cdot 1)$

Step 5.2.3

Multiply $- 1$ by $1$ .

$4 (2 - 1)$

Step 5.2.4

Subtract $1$ from $2$ .

$4 \cdot 1$

Step 5.2.5

Multiply $4$ by $1$ .

$4$

Step 6