Calculus Examples

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

Step 1

Let $u = x + 4$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

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

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Step 1.1.1

Differentiate $x + 4$ .

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

Step 1.1.2

By the Sum Rule, the derivative of $x + 4$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [4]$ .

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

Step 1.1.3

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

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

Step 1.1.4

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$1 + 0$

Step 1.1.5

Add $1$ and $0$ .

$1$

Step 1.2

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Add $1$ and $2$ .

$\int u^{3} - 4 u^{2} d u$

Step 3

Split the single integral into multiple integrals.

$\int u^{3} d u + \int - 4 u^{2} d u$

Step 4

By the Power Rule, the integral of $u^{3}$ with respect to $u$ is $\frac{1}{4} u^{4}$ .

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

Step 5

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

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

Step 6

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

$\frac{1}{4} u^{4} + C - 4 (\frac{1}{3} u^{3} + C)$

Step 7

Simplify.

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Step 7.1

Simplify.

$\frac{1}{4} u^{4} - 4 (\frac{1}{3}) u^{3} + C$

Step 7.2

Simplify.

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Step 7.2.1

Combine $- 4$ and $\frac{1}{3}$ .

$\frac{1}{4} u^{4} + \frac{- 4}{3} u^{3} + C$

Step 7.2.2

Move the negative in front of the fraction.

$\frac{1}{4} u^{4} - \frac{4}{3} u^{3} + C$

Step 8

Replace all occurrences of $u$ with $x + 4$ .

$\frac{1}{4} {(x + 4)}^{4} - \frac{4}{3} {(x + 4)}^{3} + C$