Calculus Examples

$\int {(2 x - 3)}^{2} d x$

Step 1

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

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Step 1.1

Let $u = 2 x - 3$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $2 x - 3$ .

$\frac{d}{d x} [2 x - 3]$

Step 1.1.2

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

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

Step 1.1.3

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 1.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

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 $n = 1$ .

$2 \cdot 1 + \frac{d}{d x} [- 3]$

Step 1.1.3.3

Multiply $2$ by $1$ .

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

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

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

$2 + 0$

Step 1.1.4.2

Add $2$ and $0$ .

$2$

Step 1.2

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

$\int u^{2} \frac{1}{2} d u$

Step 2

Combine $u^{2}$ and $\frac{1}{2}$ .

$\int \frac{u^{2}}{2} d u$

Step 3

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int u^{2} d u$

Step 4

By the Power Rule, the integral of $u^{2}$ with respect to $u$ is $\frac{1}{3} u^{3}$ .

$\frac{1}{2} (\frac{1}{3} u^{3} + C)$

Step 5

Simplify.

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Step 5.1

Rewrite $\frac{1}{2} (\frac{1}{3} u^{3} + C)$ as $\frac{1}{2} \cdot \frac{1}{3} u^{3} + C$ .

$\frac{1}{2} \cdot \frac{1}{3} u^{3} + C$

Step 5.2

Simplify.

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Step 5.2.1

Multiply $\frac{1}{2}$ by $\frac{1}{3}$ .

$\frac{1}{2 \cdot 3} u^{3} + C$

Step 5.2.2

Multiply $2$ by $3$ .

$\frac{1}{6} u^{3} + C$

Step 6

Replace all occurrences of $u$ with $2 x - 3$ .

$\frac{1}{6} {(2 x - 3)}^{3} + C$