Calculus Examples

$4 x^{3} {(3 x^{4} + 3)}^{9}$

Step 1

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

$4 \int x^{3} {(3 x^{4} + 3)}^{9} d x$

Step 2

Let $u = 3 x^{4} + 3$ . Then $d u = 12 x^{3} d x$ , so $\frac{1}{12} d u = x^{3} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

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

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Step 2.1.1

Differentiate $3 x^{4} + 3$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

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Step 2.1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{4}$ with respect to $x$ is $3 \frac{d}{d x} [x^{4}]$ .

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

Step 2.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

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

Step 2.1.3.3

Multiply $4$ by $3$ .

$12 x^{3} + \frac{d}{d x} [3]$

Step 2.1.4

Differentiate using the Constant Rule.

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Step 2.1.4.1

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$12 x^{3} + 0$

Step 2.1.4.2

Add $12 x^{3}$ and $0$ .

$12 x^{3}$

Step 2.2

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

$4 \int u^{9} \frac{1}{12} d u$

Step 3

Combine $u^{9}$ and $\frac{1}{12}$ .

$4 \int \frac{u^{9}}{12} d u$

Step 4

Since $\frac{1}{12}$ is constant with respect to $u$ , move $\frac{1}{12}$ out of the integral.

$4 (\frac{1}{12} \int u^{9} d u)$

Step 5

Simplify.

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Step 5.1

Combine $\frac{1}{12}$ and $4$ .

$\frac{4}{12} \int u^{9} d u$

Step 5.2

Cancel the common factor of $4$ and $12$ .

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Step 5.2.1

Factor $4$ out of $4$ .

$\frac{4 (1)}{12} \int u^{9} d u$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $4$ out of $12$ .

$\frac{4 \cdot 1}{4 \cdot 3} \int u^{9} d u$

Step 5.2.2.2

Cancel the common factor.

$\frac{4 \cdot 1}{4 \cdot 3} \int u^{9} d u$

Step 5.2.2.3

Rewrite the expression.

$\frac{1}{3} \int u^{9} d u$

Step 6

By the Power Rule, the integral of $u^{9}$ with respect to $u$ is $\frac{1}{10} u^{10}$ .

$\frac{1}{3} (\frac{1}{10} u^{10} + C)$

Step 7

Simplify.

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Step 7.1

Rewrite $\frac{1}{3} (\frac{1}{10} u^{10} + C)$ as $\frac{1}{3} \cdot \frac{1}{10} u^{10} + C$ .

$\frac{1}{3} \cdot \frac{1}{10} u^{10} + C$

Step 7.2

Simplify.

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Step 7.2.1

Multiply $\frac{1}{3}$ by $\frac{1}{10}$ .

$\frac{1}{3 \cdot 10} u^{10} + C$

Step 7.2.2

Multiply $3$ by $10$ .

$\frac{1}{30} u^{10} + C$

Step 8

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

$\frac{1}{30} {(3 x^{4} + 3)}^{10} + C$