Calculus Examples

$\int \frac{3}{\sqrt[4]{2 x + 5}} d x$

Step 1

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

$3 \int \frac{1}{\sqrt[4]{2 x + 5}} d x$

Step 2

Let $u = 2 x + 5$ . 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 $u = 2 x + 5$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $2 x + 5$ .

$\frac{d}{d x} [2 x + 5]$

Step 2.1.2

By the Sum Rule, the derivative of $2 x + 5$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [2 x] + \frac{d}{d x} [5]$

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

$2 \frac{d}{d x} [x] + \frac{d}{d x} [5]$

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 $n = 1$ .

$2 \cdot 1 + \frac{d}{d x} [5]$

Step 2.1.3.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [5]$

Step 2.1.4

Differentiate using the Constant Rule.

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Step 2.1.4.1

Since $5$ is constant with respect to $x$ , the derivative of $5$ 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 $d u$ .

$3 \int \frac{1}{\sqrt[4]{u}} \cdot \frac{1}{2} d u$

Step 3

Simplify.

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Step 3.1

Multiply $\frac{1}{\sqrt[4]{u}}$ by $\frac{1}{2}$ .

$3 \int \frac{1}{\sqrt[4]{u} \cdot 2} d u$

Step 3.2

Move $2$ to the left of $\sqrt[4]{u}$ .

$3 \int \frac{1}{2 \sqrt[4]{u}} d u$

Step 4

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

$3 (\frac{1}{2} \int \frac{1}{\sqrt[4]{u}} d u)$

Step 5

Combine fractions.

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Step 5.1

Combine $\frac{1}{2}$ and $3$ .

$\frac{3}{2} \int \frac{1}{\sqrt[4]{u}} d u$

Step 5.2

Apply basic rules of exponents.

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Step 5.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{u}$ as $u^{\frac{1}{4}}$ .

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

Step 5.2.2

Move $u^{\frac{1}{4}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 5.2.3

Multiply the exponents in ${(u^{\frac{1}{4}})}^{- 1}$ .

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Step 5.2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 5.2.3.2

Combine $\frac{1}{4}$ and $- 1$ .

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

Step 5.2.3.3

Move the negative in front of the fraction.

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Rewrite $\frac{3}{2} (\frac{4}{3} u^{\frac{3}{4}} + C)$ as $\frac{3}{2} \cdot \frac{4}{3} u^{\frac{3}{4}} + C$ .

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

Step 7.2

Simplify.

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Step 7.2.1

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

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

Step 7.2.2

Multiply $3$ by $4$ .

$\frac{12}{2 \cdot 3} u^{\frac{3}{4}} + C$

Step 7.2.3

Multiply $2$ by $3$ .

$\frac{12}{6} u^{\frac{3}{4}} + C$

Step 7.2.4

Cancel the common factor of $12$ and $6$ .

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Step 7.2.4.1

Factor $6$ out of $12$ .

$\frac{6 \cdot 2}{6} u^{\frac{3}{4}} + C$

Step 7.2.4.2

Cancel the common factors.

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Step 7.2.4.2.1

Factor $6$ out of $6$ .

$\frac{6 \cdot 2}{6 (1)} u^{\frac{3}{4}} + C$

Step 7.2.4.2.2

Cancel the common factor.

$\frac{6 \cdot 2}{6 \cdot 1} u^{\frac{3}{4}} + C$

Step 7.2.4.2.3

Rewrite the expression.

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

Step 7.2.4.2.4

Divide $2$ by $1$ .

$2 u^{\frac{3}{4}} + C$

Step 8

Replace all occurrences of $u$ with $2 x + 5$ .

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