Calculus Examples

$\int x {(x + 2)}^{4} d x$

Step 1

This integral could not be completed using integration by parts. Mathway will use another method.

Step 2

Let $u = x + 2$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

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

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Step 2.1.1

Differentiate $x + 2$ .

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

Step 2.1.2

By the Sum Rule, the derivative of $x + 2$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [2]$ .

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

Step 2.1.3

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

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

Step 2.1.4

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$1 + 0$

Step 2.1.5

Add $1$ and $0$ .

$1$

Step 2.2

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

$\int (u - 2) u^{4} d u$

Step 3

Multiply $(u - 2) u^{4}$ .

$\int u \cdot u^{4} - 2 u^{4} d u$

Step 4

Multiply $u$ by $u^{4}$ by adding the exponents.

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Step 4.1

Multiply $u$ by $u^{4}$ .

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Step 4.1.1

Raise $u$ to the power of $1$ .

$\int u^{1} u^{4} - 2 u^{4} d u$

Step 4.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int u^{1 + 4} - 2 u^{4} d u$

Step 4.2

Add $1$ and $4$ .

$\int u^{5} - 2 u^{4} d u$

Step 5

Split the single integral into multiple integrals.

$\int u^{5} d u + \int - 2 u^{4} d u$

Step 6

By the Power Rule, the integral of $u^{5}$ with respect to $u$ is $\frac{1}{6} u^{6}$ .

$\frac{1}{6} u^{6} + C + \int - 2 u^{4} d u$

Step 7

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

$\frac{1}{6} u^{6} + C - 2 \int u^{4} d u$

Step 8

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

$\frac{1}{6} u^{6} + C - 2 (\frac{1}{5} u^{5} + C)$

Step 9

Simplify.

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Step 9.1

Simplify.

$\frac{1}{6} u^{6} - 2 (\frac{1}{5}) u^{5} + C$

Step 9.2

Simplify.

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Step 9.2.1

Combine $- 2$ and $\frac{1}{5}$ .

$\frac{1}{6} u^{6} + \frac{- 2}{5} u^{5} + C$

Step 9.2.2

Move the negative in front of the fraction.

$\frac{1}{6} u^{6} - \frac{2}{5} u^{5} + C$

Step 10

Replace all occurrences of $u$ with $x + 2$ .

$\frac{1}{6} {(x + 2)}^{6} - \frac{2}{5} {(x + 2)}^{5} + C$