Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $5 + x$ .

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

Step 1.1.2

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 1.1.3

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 1.1.4

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 1.1.5

Add $0$ and $1$ .

$1$

Step 1.2

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

$\int u^{- 5} d u$

Step 2

By the Power Rule, the integral of $u^{- 5}$ with respect to $u$ is $- \frac{1}{4} u^{- 4}$ .

$- \frac{1}{4} u^{- 4} + C$

Step 3

Simplify.

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Step 3.1

Rewrite $- \frac{1}{4} u^{- 4} + C$ as $- \frac{1}{4} \cdot \frac{1}{u^{4}} + C$ .

$- \frac{1}{4} \cdot \frac{1}{u^{4}} + C$

Step 3.2

Simplify.

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Step 3.2.1

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

$- \frac{1}{u^{4} \cdot 4} + C$

Step 3.2.2

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

$- \frac{1}{4 u^{4}} + C$

Step 4

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

$- \frac{1}{4 {(5 + x)}^{4}} + C$