Calculus Examples

$\frac{9 x^{2}}{\sqrt{1 - x^{3}}}$

Step 1

Write $\frac{9 x^{2}}{\sqrt{1 - x^{3}}}$ as a function.

$f (x) = \frac{9 x^{2}}{\sqrt{1 - x^{3}}}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \frac{9 x^{2}}{\sqrt{1 - x^{3}}} d x$

Step 4

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

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

Step 5

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

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Step 5.1

Let $u = 1 - x^{3}$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $1 - x^{3}$ .

$\frac{d}{d x} [1 - x^{3}]$

Step 5.1.2

Differentiate.

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Step 5.1.2.1

By the Sum Rule, the derivative of $1 - x^{3}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- x^{3}]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [- x^{3}]$

Step 5.1.2.2

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

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

Step 5.1.3

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

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Step 5.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{3}$ with respect to $x$ is $- \frac{d}{d x} [x^{3}]$ .

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

Step 5.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 5.1.3.3

Multiply $3$ by $- 1$ .

$0 - 3 x^{2}$

Step 5.1.4

Subtract $3 x^{2}$ from $0$ .

$- 3 x^{2}$

Step 5.2

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

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

Step 6

Simplify.

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Step 6.1

Move the negative in front of the fraction.

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

Step 6.2

Multiply $\frac{1}{\sqrt{u}}$ by $\frac{1}{3}$ .

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

Step 6.3

Move $3$ to the left of $\sqrt{u}$ .

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

Step 7

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

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

Step 8

Multiply $- 1$ by $9$ .

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

Step 9

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

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

Step 10

Simplify the expression.

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Step 10.1

Simplify.

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Step 10.1.1

Combine $\frac{1}{3}$ and $- 9$ .

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

Step 10.1.2

Cancel the common factor of $- 9$ and $3$ .

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Step 10.1.2.1

Factor $3$ out of $- 9$ .

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

Step 10.1.2.2

Cancel the common factors.

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Step 10.1.2.2.1

Factor $3$ out of $3$ .

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

Step 10.1.2.2.2

Cancel the common factor.

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

Step 10.1.2.2.3

Rewrite the expression.

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

Step 10.1.2.2.4

Divide $- 3$ by $1$ .

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

Step 10.2

Apply basic rules of exponents.

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Step 10.2.1

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

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

Step 10.2.2

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

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

Step 10.2.3

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

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Step 10.2.3.1

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

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

Step 10.2.3.2

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

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

Step 10.2.3.3

Move the negative in front of the fraction.

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

Step 11

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

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

Step 12

Simplify.

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Step 12.1

Rewrite $- 3 (2 u^{\frac{1}{2}} + C)$ as $- 3 \cdot 2 u^{\frac{1}{2}} + C$ .

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

Step 12.2

Multiply $- 3$ by $2$ .

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

Step 13

Replace all occurrences of $u$ with $1 - x^{3}$ .

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

Step 14

The answer is the antiderivative of the function $f (x) = \frac{9 x^{2}}{\sqrt{1 - x^{3}}}$ .

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