Calculus Examples

$\int (x + 5) e^{{(x + 5)}^{2}} d x$

Step 1

Let $u_{1} = x + 5$ . Then $d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 1.1

Let $u_{1} = x + 5$ . Find $\frac{d u_{1}}{d x}$ .

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Step 1.1.1

Differentiate $x + 5$ .

$\frac{d}{d x} [x + 5]$

Step 1.1.2

By the Sum Rule, the derivative of $x + 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [5]$

Step 1.1.3

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

$1 + \frac{d}{d x} [5]$

Step 1.1.4

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$1 + 0$

Step 1.1.5

Add $1$ and $0$ .

$1$

Step 1.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int u_{1} e^{{u_{1}}^{2}} d u_{1}$

Step 2

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

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Step 2.1

Let $u_{2} = {u_{1}}^{2}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 2.1.1

Differentiate ${u_{1}}^{2}$ .

$\frac{d}{d u_{1}} [{u_{1}}^{2}]$

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 2$ .

$2 u_{1}$

Step 2.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\int e^{u_{2}} \frac{1}{2} d u_{2}$

Step 3

Combine $e^{u_{2}}$ and $\frac{1}{2}$ .

$\int \frac{e^{u_{2}}}{2} d u_{2}$

Step 4

Since $\frac{1}{2}$ is constant with respect to $u_{2}$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int e^{u_{2}} d u_{2}$

Step 5

The integral of $e^{u_{2}}$ with respect to $u_{2}$ is $e^{u_{2}}$ .

$\frac{1}{2} (e^{u_{2}} + C)$

Step 6

Simplify.

$\frac{1}{2} e^{u_{2}} + C$

Step 7

Substitute back in for each integration substitution variable.

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Step 7.1

Replace all occurrences of $u_{2}$ with ${u_{1}}^{2}$ .

$\frac{1}{2} e^{{u_{1}}^{2}} + C$

Step 7.2

Replace all occurrences of $u_{1}$ with $x + 5$ .

$\frac{1}{2} e^{{(x + 5)}^{2}} + C$