Calculus Examples

$\frac{x - x^{2}}{2 \sqrt[3]{x}}$

Step 1

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

$f (x) = \frac{x - x^{2}}{2 \sqrt[3]{x}}$

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.

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

Step 4

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

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

Step 5

Simplify the expression.

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Step 5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

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

Step 5.2

Simplify.

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Step 5.2.1

Factor $x$ out of $x - x^{2}$ .

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Step 5.2.1.1

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

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

Step 5.2.1.2

Factor $x$ out of $x^{1}$ .

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

Step 5.2.1.3

Factor $x$ out of $- x^{2}$ .

$\frac{1}{2} \int \frac{x \cdot 1 + x (- x)}{x^{\frac{1}{3}}} d x$

Step 5.2.1.4

Factor $x$ out of $x \cdot 1 + x (- x)$ .

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

Step 5.2.2

Move $x^{\frac{1}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 5.2.3

Multiply $x$ by $x^{- \frac{1}{3}}$ by adding the exponents.

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Step 5.2.3.1

Move $x^{- \frac{1}{3}}$ .

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

Step 5.2.3.2

Multiply $x^{- \frac{1}{3}}$ by $x$ .

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Step 5.2.3.2.1

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

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

Step 5.2.3.2.2

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

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

Step 5.2.3.3

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

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

Step 5.2.3.4

Combine the numerators over the common denominator.

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

Step 5.2.3.5

Add $- 1$ and $3$ .

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

Step 6

Expand $x^{\frac{2}{3}} (1 - x)$ .

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Step 6.1

Apply the distributive property.

$\frac{1}{2} \int x^{\frac{2}{3}} \cdot 1 + x^{\frac{2}{3}} (- x) d x$

Step 6.2

Reorder $x^{\frac{2}{3}}$ and $1$ .

$\frac{1}{2} \int 1 \cdot x^{\frac{2}{3}} + x^{\frac{2}{3}} (- x) d x$

Step 6.3

Reorder $x^{\frac{2}{3}}$ and $- 1$ .

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

Step 6.4

Multiply $x^{\frac{2}{3}}$ by $1$ .

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

Step 6.5

Factor out negative.

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

Step 6.6

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

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

Step 6.7

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

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

Step 6.8

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

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

Step 6.9

Combine the numerators over the common denominator.

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

Step 6.10

Add $2$ and $3$ .

$\frac{1}{2} \int x^{\frac{2}{3}} - x^{\frac{5}{3}} d x$

Step 7

Split the single integral into multiple integrals.

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

Step 8

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

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

Step 9

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

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

Step 10

By the Power Rule, the integral of $x^{\frac{5}{3}}$ with respect to $x$ is $\frac{3}{8} x^{\frac{8}{3}}$ .

$\frac{1}{2} (\frac{3}{5} x^{\frac{5}{3}} + C - (\frac{3}{8} x^{\frac{8}{3}} + C))$

Step 11

Simplify.

$\frac{1}{2} (\frac{3}{5} x^{\frac{5}{3}} - \frac{3}{8} x^{\frac{8}{3}}) + C$

Step 12

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

$F (x) =$ $\frac{1}{2} (\frac{3}{5} x^{\frac{5}{3}} - \frac{3}{8} x^{\frac{8}{3}}) + C$