Calculus Examples

$\int \frac{3 x^{2}}{2 \sqrt{5 - 3 x^{3}}} d x$

Step 1

Since $\frac{3}{2}$ is constant with respect to $x$ , move $\frac{3}{2}$ out of the integral.

$\frac{3}{2} \int \frac{x^{2}}{\sqrt{5 - 3 x^{3}}} d x$

Step 2

Let $u = 5 - 3 x^{3}$ . Then $d u = - 9 x^{2} d x$ , so $- \frac{1}{9} d u = x^{2} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 5 - 3 x^{3}$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $5 - 3 x^{3}$ .

$\frac{d}{d x} [5 - 3 x^{3}]$

Step 2.1.2

Differentiate.

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Step 2.1.2.1

By the Sum Rule, the derivative of $5 - 3 x^{3}$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [- 3 x^{3}]$ .

$\frac{d}{d x} [5] + \frac{d}{d x} [- 3 x^{3}]$

Step 2.1.2.2

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 3 x^{3}]$

Step 2.1.3

Evaluate $\frac{d}{d x} [- 3 x^{3}]$ .

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Step 2.1.3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{3}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{3}]$ .

$0 - 3 \frac{d}{d x} [x^{3}]$

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 $n = 3$ .

$0 - 3 (3 x^{2})$

Step 2.1.3.3

Multiply $3$ by $- 3$ .

$0 - 9 x^{2}$

Step 2.1.4

Subtract $9 x^{2}$ from $0$ .

$- 9 x^{2}$

Step 2.2

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

$\frac{3}{2} \int \frac{1}{\sqrt{u}} \cdot \frac{1}{- 9} d u$

Step 3

Simplify.

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Step 3.1

Move the negative in front of the fraction.

$\frac{3}{2} \int \frac{1}{\sqrt{u}} (- \frac{1}{9}) d u$

Step 3.2

Multiply $\frac{1}{\sqrt{u}}$ by $\frac{1}{9}$ .

$\frac{3}{2} \int - \frac{1}{\sqrt{u} \cdot 9} d u$

Step 3.3

Move $9$ to the left of $\sqrt{u}$ .

$\frac{3}{2} \int - \frac{1}{9 \sqrt{u}} d u$

Step 4

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

$\frac{3}{2} (- \int \frac{1}{9 \sqrt{u}} d u)$

Step 5

Since $\frac{1}{9}$ is constant with respect to $u$ , move $\frac{1}{9}$ out of the integral.

$\frac{3}{2} (- (\frac{1}{9} \int \frac{1}{\sqrt{u}} d u))$

Step 6

Simplify the expression.

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Step 6.1

Simplify.

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Step 6.1.1

Multiply $\frac{3}{2}$ by $\frac{1}{9}$ .

$- \frac{3}{2 \cdot 9} \int \frac{1}{\sqrt{u}} d u$

Step 6.1.2

Multiply $2$ by $9$ .

$- \frac{3}{18} \int \frac{1}{\sqrt{u}} d u$

Step 6.1.3

Cancel the common factor of $3$ and $18$ .

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Step 6.1.3.1

Factor $3$ out of $3$ .

$- \frac{3 (1)}{18} \int \frac{1}{\sqrt{u}} d u$

Step 6.1.3.2

Cancel the common factors.

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Step 6.1.3.2.1

Factor $3$ out of $18$ .

$- \frac{3 \cdot 1}{3 \cdot 6} \int \frac{1}{\sqrt{u}} d u$

Step 6.1.3.2.2

Cancel the common factor.

$- \frac{3 \cdot 1}{3 \cdot 6} \int \frac{1}{\sqrt{u}} d u$

Step 6.1.3.2.3

Rewrite the expression.

$- \frac{1}{6} \int \frac{1}{\sqrt{u}} d u$

Step 6.2

Apply basic rules of exponents.

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Step 6.2.1

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

$- \frac{1}{6} \int \frac{1}{u^{\frac{1}{2}}} d u$

Step 6.2.2

Move $u^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$- \frac{1}{6} \int {(u^{\frac{1}{2}})}^{- 1} d u$

Step 6.2.3

Multiply the exponents in ${(u^{\frac{1}{2}})}^{- 1}$ .

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Step 6.2.3.1

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

$- \frac{1}{6} \int u^{\frac{1}{2} \cdot - 1} d u$

Step 6.2.3.2

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

$- \frac{1}{6} \int u^{\frac{- 1}{2}} d u$

Step 6.2.3.3

Move the negative in front of the fraction.

$- \frac{1}{6} \int u^{- \frac{1}{2}} d u$

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Rewrite $- \frac{1}{6} (2 u^{\frac{1}{2}} + C)$ as $- \frac{1}{6} \cdot 2 u^{\frac{1}{2}} + C$ .

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

Step 8.2

Simplify.

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Step 8.2.1

Multiply $2$ by $- 1$ .

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

Step 8.2.2

Combine $- 2$ and $\frac{1}{6}$ .

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

Step 8.2.3

Cancel the common factor of $- 2$ and $6$ .

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Step 8.2.3.1

Factor $2$ out of $- 2$ .

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

Step 8.2.3.2

Cancel the common factors.

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Step 8.2.3.2.1

Factor $2$ out of $6$ .

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

Step 8.2.3.2.2

Cancel the common factor.

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

Step 8.2.3.2.3

Rewrite the expression.

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

Step 8.2.4

Move the negative in front of the fraction.

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

Step 9

Replace all occurrences of $u$ with $5 - 3 x^{3}$ .

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