Calculus Examples

$\int {(3 x^{4} + 2)}^{4} (12 x^{3}) d x$

Step 1

Move $12$ to the left of ${(3 x^{4} + 2)}^{4}$ .

$\int 12 {(3 x^{4} + 2)}^{4} x^{3} d x$

Step 2

Let $u = 3 x^{4} + 2$ . Then $d u = 12 x^{3} d x$ , so $\frac{1}{12 x^{3}} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 3 x^{4} + 2$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $3 x^{4} + 2$ .

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

Step 2.1.2

By the Sum Rule, the derivative of $3 x^{4} + 2$ with respect to $x$ is $\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [2]$ .

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

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

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

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 $n = 4$ .

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

Step 2.1.3.3

Multiply $4$ by $3$ .

$12 x^{3} + \frac{d}{d x} [2]$

Step 2.1.4

Differentiate using the Constant Rule.

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Step 2.1.4.1

Since $2$ is constant with respect to $x$ , the derivative of $2$ 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 $d u$ .

$\int u^{4} d u$

Step 3

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

$\frac{1}{5} u^{5} + C$

Step 4

Replace all occurrences of $u$ with $3 x^{4} + 2$ .

$\frac{1}{5} {(3 x^{4} + 2)}^{5} + C$