Calculus Examples

$\int \frac{9}{2 {(9 - 2 x)}^{\frac{1}{4}}} d x$

Step 1

Since $\frac{9}{2}$ is constant with respect to $x$ , move $\frac{9}{2}$ out of the integral.

$\frac{9}{2} \int \frac{1}{{(9 - 2 x)}^{\frac{1}{4}}} d x$

Step 2

Let $u = 9 - 2 x$ . 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 $u = 9 - 2 x$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $9 - 2 x$ .

$\frac{d}{d x} [9 - 2 x]$

Step 2.1.2

Differentiate.

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Step 2.1.2.1

By the Sum Rule, the derivative of $9 - 2 x$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [- 2 x]$ .

$\frac{d}{d x} [9] + \frac{d}{d x} [- 2 x]$

Step 2.1.2.2

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 2 x]$

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

$0 - 2 \frac{d}{d x} [x]$

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 $n = 1$ .

$0 - 2 \cdot 1$

Step 2.1.3.3

Multiply $- 2$ by $1$ .

$0 - 2$

Step 2.1.4

Subtract $2$ from $0$ .

$- 2$

Step 2.2

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

$\frac{9}{2} \int \frac{1}{u^{\frac{1}{4}}} \cdot \frac{1}{- 2} d u$

Step 3

Simplify.

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Step 3.1

Move the negative in front of the fraction.

$\frac{9}{2} \int \frac{1}{u^{\frac{1}{4}}} (- \frac{1}{2}) d u$

Step 3.2

Multiply $\frac{1}{u^{\frac{1}{4}}}$ by $\frac{1}{2}$ .

$\frac{9}{2} \int - \frac{1}{u^{\frac{1}{4}} \cdot 2} d u$

Step 3.3

Move $2$ to the left of $u^{\frac{1}{4}}$ .

$\frac{9}{2} \int - \frac{1}{2 u^{\frac{1}{4}}} d u$

Step 4

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

$\frac{9}{2} (- \int \frac{1}{2 u^{\frac{1}{4}}} d u)$

Step 5

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

$\frac{9}{2} (- (\frac{1}{2} \int \frac{1}{u^{\frac{1}{4}}} d u))$

Step 6

Simplify the expression.

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Step 6.1

Simplify.

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Step 6.1.1

Multiply $\frac{9}{2}$ by $\frac{1}{2}$ .

$- \frac{9}{2 \cdot 2} \int \frac{1}{u^{\frac{1}{4}}} d u$

Step 6.1.2

Multiply $2$ by $2$ .

$- \frac{9}{4} \int \frac{1}{u^{\frac{1}{4}}} d u$

Step 6.2

Apply basic rules of exponents.

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Step 6.2.1

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

$- \frac{9}{4} \int {(u^{\frac{1}{4}})}^{- 1} d u$

Step 6.2.2

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

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Step 6.2.2.1

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

$- \frac{9}{4} \int u^{\frac{1}{4} \cdot - 1} d u$

Step 6.2.2.2

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

$- \frac{9}{4} \int u^{\frac{- 1}{4}} d u$

Step 6.2.2.3

Move the negative in front of the fraction.

$- \frac{9}{4} \int u^{- \frac{1}{4}} d u$

Step 7

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

$- \frac{9}{4} (\frac{4}{3} u^{\frac{3}{4}} + C)$

Step 8

Simplify.

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Step 8.1

Rewrite $- \frac{9}{4} (\frac{4}{3} u^{\frac{3}{4}} + C)$ as $- \frac{9}{4} \cdot \frac{4}{3} u^{\frac{3}{4}} + C$ .

$- \frac{9}{4} \cdot \frac{4}{3} u^{\frac{3}{4}} + C$

Step 8.2

Simplify.

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Step 8.2.1

Multiply $\frac{4}{3}$ by $\frac{9}{4}$ .

$- \frac{4 \cdot 9}{3 \cdot 4} u^{\frac{3}{4}} + C$

Step 8.2.2

Multiply $4$ by $9$ .

$- \frac{36}{3 \cdot 4} u^{\frac{3}{4}} + C$

Step 8.2.3

Multiply $3$ by $4$ .

$- \frac{36}{12} u^{\frac{3}{4}} + C$

Step 8.2.4

Cancel the common factor of $36$ and $12$ .

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Step 8.2.4.1

Factor $12$ out of $36$ .

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

Step 8.2.4.2

Cancel the common factors.

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Step 8.2.4.2.1

Factor $12$ out of $12$ .

$- \frac{12 \cdot 3}{12 (1)} u^{\frac{3}{4}} + C$

Step 8.2.4.2.2

Cancel the common factor.

$- \frac{12 \cdot 3}{12 \cdot 1} u^{\frac{3}{4}} + C$

Step 8.2.4.2.3

Rewrite the expression.

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

Step 8.2.4.2.4

Divide $3$ by $1$ .

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

Step 8.2.5

Multiply $- 1$ by $3$ .

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

Step 9

Replace all occurrences of $u$ with $9 - 2 x$ .

$- 3 {(9 - 2 x)}^{\frac{3}{4}} + C$