Calculus Examples

$\int 3 {(3 x - 1)}^{4} d x$

Step 1

Let $u = 3 x - 1$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 3 x - 1$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $3 x - 1$ .

$\frac{d}{d x} [3 x - 1]$

Step 1.1.2

By the Sum Rule, the derivative of $3 x - 1$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1]$

Step 1.1.3

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

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Step 1.1.3.1

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

$3 \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

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 $n = 1$ .

$3 \cdot 1 + \frac{d}{d x} [- 1]$

Step 1.1.3.3

Multiply $3$ by $1$ .

$3 + \frac{d}{d x} [- 1]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$3 + 0$

Step 1.1.4.2

Add $3$ and $0$ .

$3$

Step 1.2

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

$\int u^{4} d u$

Step 2

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 3

Replace all occurrences of $u$ with $3 x - 1$ .

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