Calculus Examples

$\int 3 {(5 + x)}^{2} d x$

Step 1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int {(5 + x)}^{2} d x$

Step 2

Let $u = 5 + x$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

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

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Step 2.1.1

Differentiate $5 + x$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

$0 + \frac{d}{d x} [x]$

Step 2.1.4

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

$0 + 1$

Step 2.1.5

Add $0$ and $1$ .

$1$

Step 2.2

Rewrite the problem using $u$ and $d u$ .

$3 \int u^{2} d u$

Step 3

By the Power Rule, the integral of $u^{2}$ with respect to $u$ is $\frac{1}{3} u^{3}$ .

$3 (\frac{1}{3} u^{3} + C)$

Step 4

Simplify.

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Step 4.1

Rewrite $3 (\frac{1}{3} u^{3} + C)$ as $3 (\frac{1}{3}) u^{3} + C$ .

$3 (\frac{1}{3}) u^{3} + C$

Step 4.2

Simplify.

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Step 4.2.1

Combine $3$ and $\frac{1}{3}$ .

$\frac{3}{3} u^{3} + C$

Step 4.2.2

Cancel the common factor of $3$ .

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Step 4.2.2.1

Cancel the common factor.

$\frac{3}{3} u^{3} + C$

Step 4.2.2.2

Rewrite the expression.

$1 u^{3} + C$

Step 4.2.3

Multiply $u^{3}$ by $1$ .

$u^{3} + C$

Step 5

Replace all occurrences of $u$ with $5 + x$ .

${(5 + x)}^{3} + C$