Calculus Examples

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

Step 1

Let $u = 4 x^{4} + 5$ . Then $d u = 16 x^{3} d x$ , so $\frac{1}{16} d u = x^{3} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

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

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Step 1.1.1

Differentiate $4 x^{4} + 5$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

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 $n = 4$ .

$4 (4 x^{3}) + \frac{d}{d x} [5]$

Step 1.1.3.3

Multiply $4$ by $4$ .

$16 x^{3} + \frac{d}{d x} [5]$

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 $0$ .

$16 x^{3} + 0$

Step 1.1.4.2

Add $16 x^{3}$ and $0$ .

$16 x^{3}$

Step 1.2

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

$\int 4 u \frac{1}{16} d u$

Step 2

Simplify.

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Step 2.1

Combine $\frac{1}{16}$ and $4$ .

$\int \frac{4}{16} u d u$

Step 2.2

Combine $\frac{4}{16}$ and $u$ .

$\int \frac{4 u}{16} d u$

Step 2.3

Cancel the common factor of $4$ and $16$ .

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Step 2.3.1

Factor $4$ out of $4 u$ .

$\int \frac{4 (u)}{16} d u$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $4$ out of $16$ .

$\int \frac{4 u}{4 \cdot 4} d u$

Step 2.3.2.2

Cancel the common factor.

$\int \frac{4 u}{4 \cdot 4} d u$

Step 2.3.2.3

Rewrite the expression.

$\int \frac{u}{4} d u$

Step 3

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int u d u$

Step 4

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

$\frac{1}{4} (\frac{1}{2} u^{2} + C)$

Step 5

Simplify.

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Step 5.1

Rewrite $\frac{1}{4} (\frac{1}{2} u^{2} + C)$ as $\frac{1}{4} \cdot \frac{1}{2} u^{2} + C$ .

$\frac{1}{4} \cdot \frac{1}{2} u^{2} + C$

Step 5.2

Simplify.

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Step 5.2.1

Multiply $\frac{1}{4}$ by $\frac{1}{2}$ .

$\frac{1}{4 \cdot 2} u^{2} + C$

Step 5.2.2

Multiply $4$ by $2$ .

$\frac{1}{8} u^{2} + C$

Step 6

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

$\frac{1}{8} {(4 x^{4} + 5)}^{2} + C$