Calculus Examples

$\int 3 x {(2 x - 1)}^{2} d x$

Step 1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int x {(2 x - 1)}^{2} d x$

Step 2

Let $u = 2 x - 1$ . 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 2.1

Let $u = 2 x - 1$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $2 x - 1$ .

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

Step 2.1.2

By the Sum Rule, the derivative of $2 x - 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]$ .

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

Step 2.1.3

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

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Step 2.1.3.1

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

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

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 $n = 1$ .

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

Step 2.1.3.3

Multiply $2$ by $1$ .

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

Step 2.1.4

Differentiate using the Constant Rule.

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Step 2.1.4.1

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

$2 + 0$

Step 2.1.4.2

Add $2$ and $0$ .

$2$

Step 2.2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $\frac{u}{2}$ by $\frac{u^{2}}{2}$ .

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

Step 4.3

Raise $u$ to the power of $1$ .

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

Step 4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.5

Add $1$ and $2$ .

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

Step 4.6

Multiply $2$ by $2$ .

$3 \int \frac{u^{3}}{4} + \frac{1}{2} \cdot \frac{u^{2}}{2} d u$

Step 4.7

Multiply $\frac{1}{2}$ by $\frac{u^{2}}{2}$ .

$3 \int \frac{u^{3}}{4} + \frac{u^{2}}{2 \cdot 2} d u$

Step 4.8

Multiply $2$ by $2$ .

$3 \int \frac{u^{3}}{4} + \frac{u^{2}}{4} d u$

Step 5

Split the single integral into multiple integrals.

$3 (\int \frac{u^{3}}{4} d u + \int \frac{u^{2}}{4} d u)$

Step 6

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$3 (\frac{1}{4} \int u^{3} d u + \int \frac{u^{2}}{4} d u)$

Step 7

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

$3 (\frac{1}{4} (\frac{1}{4} u^{4} + C) + \int \frac{u^{2}}{4} d u)$

Step 8

Combine $\frac{1}{4}$ and $u^{4}$ .

$3 (\frac{1}{4} (\frac{u^{4}}{4} + C) + \int \frac{u^{2}}{4} d u)$

Step 9

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$3 (\frac{1}{4} (\frac{u^{4}}{4} + C) + \frac{1}{4} \int u^{2} d u)$

Step 10

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

$3 (\frac{1}{4} (\frac{u^{4}}{4} + C) + \frac{1}{4} (\frac{1}{3} u^{3} + C))$

Step 11

Simplify.

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Step 11.1

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

$3 (\frac{1}{4} (\frac{u^{4}}{4} + C) + \frac{1}{4} (\frac{u^{3}}{3} + C))$

Step 11.2

Simplify.

$3 (\frac{u^{4}}{16} + \frac{u^{3}}{12}) + C$

Step 12

Replace all occurrences of $u$ with $2 x - 1$ .

$3 (\frac{{(2 x - 1)}^{4}}{16} + \frac{{(2 x - 1)}^{3}}{12}) + C$

Step 13

Reorder terms.

$3 (\frac{1}{16} {(2 x - 1)}^{4} + \frac{1}{12} {(2 x - 1)}^{3}) + C$