Calculus Examples

$\int 48 x^{2} {(x^{3} + 4)}^{3} d x$

Step 1

Since $48$ is constant with respect to $x$ , move $48$ out of the integral.

$48 \int x^{2} {(x^{3} + 4)}^{3} d x$

Step 2

Let $u = x^{3} + 4$ . Then $d u = 3 x^{2} d x$ , so $\frac{1}{3} d u = x^{2} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = x^{3} + 4$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $x^{3} + 4$ .

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

Step 2.1.2

By the Sum Rule, the derivative of $x^{3} + 4$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [4]$ .

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

Step 2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2} + \frac{d}{d x} [4]$

Step 2.1.4

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

$3 x^{2} + 0$

Step 2.1.5

Add $3 x^{2}$ and $0$ .

$3 x^{2}$

Step 2.2

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

$48 \int u^{3} \frac{1}{3} d u$

Step 3

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

$48 \int \frac{u^{3}}{3} d u$

Step 4

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$48 (\frac{1}{3} \int u^{3} d u)$

Step 5

Simplify.

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Step 5.1

Combine $\frac{1}{3}$ and $48$ .

$\frac{48}{3} \int u^{3} d u$

Step 5.2

Cancel the common factor of $48$ and $3$ .

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Step 5.2.1

Factor $3$ out of $48$ .

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

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 16}{3 (1)} \int u^{3} d u$

Step 5.2.2.2

Cancel the common factor.

$\frac{3 \cdot 16}{3 \cdot 1} \int u^{3} d u$

Step 5.2.2.3

Rewrite the expression.

$\frac{16}{1} \int u^{3} d u$

Step 5.2.2.4

Divide $16$ by $1$ .

$16 \int u^{3} d u$

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Rewrite $16 (\frac{1}{4} u^{4} + C)$ as $16 (\frac{1}{4}) u^{4} + C$ .

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

Step 7.2

Simplify.

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Step 7.2.1

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

$\frac{16}{4} u^{4} + C$

Step 7.2.2

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

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Step 7.2.2.1

Factor $4$ out of $16$ .

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

Step 7.2.2.2

Cancel the common factors.

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Step 7.2.2.2.1

Factor $4$ out of $4$ .

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

Step 7.2.2.2.2

Cancel the common factor.

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

Step 7.2.2.2.3

Rewrite the expression.

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

Step 7.2.2.2.4

Divide $4$ by $1$ .

$4 u^{4} + C$

Step 8

Replace all occurrences of $u$ with $x^{3} + 4$ .

$4 {(x^{3} + 4)}^{4} + C$