Calculus Examples

$\int x \sqrt{2 x + 1} d x$

Step 1

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 1.1

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

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Step 1.1.1

Differentiate $2 x + 1$ .

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

Step 1.1.2

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

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

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 $n = 1$ .

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

Step 1.1.3.3

Multiply $2$ by $1$ .

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

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 $0$ .

$2 + 0$

Step 1.1.4.2

Add $2$ and $0$ .

$2$

Step 1.2

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

$\int (\frac{u}{2} - \frac{1}{2}) \sqrt{u} \frac{1}{2} d u$

Step 2

Combine fractions.

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Step 2.1

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

$\int (\frac{u}{2} - \frac{1}{2}) \frac{\sqrt{u}}{2} d u$

Step 2.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Write $1$ as a fraction with a common denominator.

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

Step 3.6

Combine the numerators over the common denominator.

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

Step 3.7

Add $2$ and $1$ .

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

Step 3.8

Multiply $2$ by $2$ .

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

Step 3.9

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

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

Step 3.10

Multiply $2$ by $2$ .

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

By the Power Rule, the integral of $u^{\frac{3}{2}}$ with respect to $u$ is $\frac{2}{5} u^{\frac{5}{2}}$ .

$\frac{1}{4} (\frac{2}{5} u^{\frac{5}{2}} + C) + \int - \frac{u^{\frac{1}{2}}}{4} d u$

Step 7

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$\frac{1}{4} (\frac{2}{5} u^{\frac{5}{2}} + C) - \int \frac{u^{\frac{1}{2}}}{4} d u$

Step 8

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

$\frac{1}{4} (\frac{2}{5} u^{\frac{5}{2}} + C) - (\frac{1}{4} \int u^{\frac{1}{2}} d u)$

Step 9

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

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

Step 10

Simplify.

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Step 10.1

Simplify.

$\frac{1}{4} \cdot \frac{2}{5} u^{\frac{5}{2}} - \frac{1}{4} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$

Step 10.2

Rewrite $\frac{1}{4} \cdot \frac{2}{5} u^{\frac{5}{2}} - \frac{1}{4} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$ as $\frac{1}{10} u^{\frac{5}{2}} - \frac{1}{4} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$ .

$\frac{1}{10} u^{\frac{5}{2}} - \frac{1}{4} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$

Step 10.3

Simplify.

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Step 10.3.1

Multiply $\frac{2}{3}$ by $\frac{1}{4}$ .

$\frac{1}{10} u^{\frac{5}{2}} - \frac{2}{3 \cdot 4} u^{\frac{3}{2}} + C$

Step 10.3.2

Multiply $3$ by $4$ .

$\frac{1}{10} u^{\frac{5}{2}} - \frac{2}{12} u^{\frac{3}{2}} + C$

Step 10.3.3

Cancel the common factor of $2$ and $12$ .

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Step 10.3.3.1

Factor $2$ out of $2$ .

$\frac{1}{10} u^{\frac{5}{2}} - \frac{2 (1)}{12} u^{\frac{3}{2}} + C$

Step 10.3.3.2

Cancel the common factors.

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Step 10.3.3.2.1

Factor $2$ out of $12$ .

$\frac{1}{10} u^{\frac{5}{2}} - \frac{2 \cdot 1}{2 \cdot 6} u^{\frac{3}{2}} + C$

Step 10.3.3.2.2

Cancel the common factor.

$\frac{1}{10} u^{\frac{5}{2}} - \frac{2 \cdot 1}{2 \cdot 6} u^{\frac{3}{2}} + C$

Step 10.3.3.2.3

Rewrite the expression.

$\frac{1}{10} u^{\frac{5}{2}} - \frac{1}{6} u^{\frac{3}{2}} + C$

Step 11

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

$\frac{1}{10} {(2 x + 1)}^{\frac{5}{2}} - \frac{1}{6} {(2 x + 1)}^{\frac{3}{2}} + C$