Calculus Examples

$\int x {(4 x + 5)}^{3} d x$

Step 1

Let $u = 4 x + 5$ . Then $d u = 4 d x$ , so $\frac{1}{4} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 4 x + 5$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $4 x + 5$ .

$\frac{d}{d x} [4 x + 5]$

Step 1.1.2

By the Sum Rule, the derivative of $4 x + 5$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [4 x] + \frac{d}{d x} [5]$

Step 1.1.3

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 1.1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x] + \frac{d}{d x} [5]$

Step 1.1.3.2

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

$4 \cdot 1 + \frac{d}{d x} [5]$

Step 1.1.3.3

Multiply $4$ by $1$ .

$4 + \frac{d}{d x} [5]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

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

$4 + 0$

Step 1.1.4.2

Add $4$ and $0$ .

$4$

Step 1.2

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

$\int (\frac{u}{4} - \frac{5}{4}) u^{3} \frac{1}{4} d u$

Step 2

Combine $\frac{1}{4}$ and $u^{3}$ .

$\int (\frac{u}{4} - \frac{5}{4}) \frac{u^{3}}{4} d u$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\int \frac{u}{4} \cdot \frac{u^{3}}{4} - \frac{5}{4} \cdot \frac{u^{3}}{4} d u$

Step 3.2

Multiply $\frac{u}{4}$ by $\frac{u^{3}}{4}$ .

$\int \frac{u \cdot u^{3}}{4 \cdot 4} - \frac{5}{4} \cdot \frac{u^{3}}{4} d u$

Step 3.3

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

$\int \frac{u^{1} u^{3}}{4 \cdot 4} - \frac{5}{4} \cdot \frac{u^{3}}{4} d u$

Step 3.4

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

$\int \frac{u^{1 + 3}}{4 \cdot 4} - \frac{5}{4} \cdot \frac{u^{3}}{4} d u$

Step 3.5

Add $1$ and $3$ .

$\int \frac{u^{4}}{4 \cdot 4} - \frac{5}{4} \cdot \frac{u^{3}}{4} d u$

Step 3.6

Multiply $4$ by $4$ .

$\int \frac{u^{4}}{16} - \frac{5}{4} \cdot \frac{u^{3}}{4} d u$

Step 3.7

Combine $- \frac{5}{4}$ and $\frac{u^{3}}{4}$ .

$\int \frac{u^{4}}{16} - \frac{5 u^{3}}{4 \cdot 4} d u$

Step 3.8

Multiply $4$ by $4$ .

$\int \frac{u^{4}}{16} - \frac{5 u^{3}}{16} d u$

Step 4

Split the single integral into multiple integrals.

$\int \frac{u^{4}}{16} d u + \int - \frac{5 u^{3}}{16} d u$

Step 5

Since $\frac{1}{16}$ is constant with respect to $u$ , move $\frac{1}{16}$ out of the integral.

$\frac{1}{16} \int u^{4} d u + \int - \frac{5 u^{3}}{16} d u$

Step 6

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

$\frac{1}{16} (\frac{1}{5} u^{5} + C) + \int - \frac{5 u^{3}}{16} d u$

Step 7

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$\frac{1}{16} (\frac{1}{5} u^{5} + C) - \int \frac{5 u^{3}}{16} d u$

Step 8

Since $\frac{5}{16}$ is constant with respect to $u$ , move $\frac{5}{16}$ out of the integral.

$\frac{1}{16} (\frac{1}{5} u^{5} + C) - (\frac{5}{16} \int u^{3} d u)$

Step 9

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

$\frac{1}{16} (\frac{1}{5} u^{5} + C) - \frac{5}{16} (\frac{1}{4} u^{4} + C)$

Step 10

Simplify.

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Step 10.1

Simplify.

$\frac{1}{16} \cdot \frac{1}{5} u^{5} - \frac{5}{16} \cdot \frac{1}{4} u^{4} + C$

Step 10.2

Rewrite $\frac{1}{16} \cdot \frac{1}{5} u^{5} - \frac{5}{16} \cdot \frac{1}{4} u^{4} + C$ as $\frac{1}{80} u^{5} - \frac{5}{16} \cdot \frac{1}{4} u^{4} + C$ .

$\frac{1}{80} u^{5} - \frac{5}{16} \cdot \frac{1}{4} u^{4} + C$

Step 10.3

Simplify.

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Step 10.3.1

Multiply $\frac{1}{4}$ by $\frac{5}{16}$ .

$\frac{1}{80} u^{5} - \frac{5}{4 \cdot 16} u^{4} + C$

Step 10.3.2

Multiply $4$ by $16$ .

$\frac{1}{80} u^{5} - \frac{5}{64} u^{4} + C$

Step 11

Replace all occurrences of $u$ with $4 x + 5$ .

$\frac{1}{80} {(4 x + 5)}^{5} - \frac{5}{64} {(4 x + 5)}^{4} + C$