Calculus Examples

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

Step 1

Apply basic rules of exponents.

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Step 1.1

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

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

Step 1.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

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

Step 2

Simplify.

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Step 2.1

Factor $x^{\frac{1}{2}}$ out of $x^{3} + x^{\frac{1}{2}}$ .

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Step 2.1.1

Factor $x^{\frac{1}{2}}$ out of $x^{3}$ .

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

Step 2.1.2

Multiply by $1$ .

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

Step 2.1.3

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} x^{\frac{5}{2}} + x^{\frac{1}{2}} \cdot 1$ .

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

Step 2.2

Move $x^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 2.3

Multiply $x^{\frac{2}{3}}$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 2.3.1

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

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

Step 2.3.2

To write $\frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.3.3

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 2.3.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.4.1

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

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

Step 2.3.4.2

Multiply $3$ by $2$ .

$\int \frac{x^{\frac{5}{2}} + 1}{x^{\frac{2 \cdot 2}{6} - \frac{1}{2} \cdot \frac{3}{3}}} d x$

Step 2.3.4.3

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

$\int \frac{x^{\frac{5}{2}} + 1}{x^{\frac{2 \cdot 2}{6} - \frac{3}{2 \cdot 3}}} d x$

Step 2.3.4.4

Multiply $2$ by $3$ .

$\int \frac{x^{\frac{5}{2}} + 1}{x^{\frac{2 \cdot 2}{6} - \frac{3}{6}}} d x$

Step 2.3.5

Combine the numerators over the common denominator.

$\int \frac{x^{\frac{5}{2}} + 1}{x^{\frac{2 \cdot 2 - 3}{6}}} d x$

Step 2.3.6

Simplify the numerator.

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Step 2.3.6.1

Multiply $2$ by $2$ .

$\int \frac{x^{\frac{5}{2}} + 1}{x^{\frac{4 - 3}{6}}} d x$

Step 2.3.6.2

Subtract $3$ from $4$ .

$\int \frac{x^{\frac{5}{2}} + 1}{x^{\frac{1}{6}}} d x$

Step 3

Apply basic rules of exponents.

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Step 3.1

Move $x^{\frac{1}{6}}$ out of the denominator by raising it to the $- 1$ power.

$\int (x^{\frac{5}{2}} + 1) {(x^{\frac{1}{6}})}^{- 1} d x$

Step 3.2

Multiply the exponents in ${(x^{\frac{1}{6}})}^{- 1}$ .

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Step 3.2.1

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

$\int (x^{\frac{5}{2}} + 1) x^{\frac{1}{6} \cdot - 1} d x$

Step 3.2.2

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

$\int (x^{\frac{5}{2}} + 1) x^{\frac{- 1}{6}} d x$

Step 3.2.3

Move the negative in front of the fraction.

$\int (x^{\frac{5}{2}} + 1) x^{- \frac{1}{6}} d x$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\int x^{\frac{5}{2}} x^{- \frac{1}{6}} + 1 x^{- \frac{1}{6}} d x$

Step 4.2

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

$\int x^{\frac{5}{2} - \frac{1}{6}} + 1 x^{- \frac{1}{6}} d x$

Step 4.3

To write $\frac{5}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\int x^{\frac{5}{2} \cdot \frac{3}{3} - \frac{1}{6}} + 1 x^{- \frac{1}{6}} d x$

Step 4.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.4.1

Multiply $\frac{5}{2}$ by $\frac{3}{3}$ .

$\int x^{\frac{5 \cdot 3}{2 \cdot 3} - \frac{1}{6}} + 1 x^{- \frac{1}{6}} d x$

Step 4.4.2

Multiply $2$ by $3$ .

$\int x^{\frac{5 \cdot 3}{6} - \frac{1}{6}} + 1 x^{- \frac{1}{6}} d x$

Step 4.5

Combine the numerators over the common denominator.

$\int x^{\frac{5 \cdot 3 - 1}{6}} + 1 x^{- \frac{1}{6}} d x$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $5$ by $3$ .

$\int x^{\frac{15 - 1}{6}} + 1 x^{- \frac{1}{6}} d x$

Step 4.6.2

Subtract $1$ from $15$ .

$\int x^{\frac{14}{6}} + 1 x^{- \frac{1}{6}} d x$

Step 4.7

Cancel the common factor of $14$ and $6$ .

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Step 4.7.1

Factor $2$ out of $14$ .

$\int x^{\frac{2 (7)}{6}} + 1 x^{- \frac{1}{6}} d x$

Step 4.7.2

Cancel the common factors.

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Step 4.7.2.1

Factor $2$ out of $6$ .

$\int x^{\frac{2 \cdot 7}{2 \cdot 3}} + 1 x^{- \frac{1}{6}} d x$

Step 4.7.2.2

Cancel the common factor.

$\int x^{\frac{2 \cdot 7}{2 \cdot 3}} + 1 x^{- \frac{1}{6}} d x$

Step 4.7.2.3

Rewrite the expression.

$\int x^{\frac{7}{3}} + 1 x^{- \frac{1}{6}} d x$

Step 4.8

Multiply $x^{- \frac{1}{6}}$ by $1$ .

$\int x^{\frac{7}{3}} + x^{- \frac{1}{6}} d x$

Step 5

Split the single integral into multiple integrals.

$\int x^{\frac{7}{3}} d x + \int x^{- \frac{1}{6}} d x$

Step 6

By the Power Rule, the integral of $x^{\frac{7}{3}}$ with respect to $x$ is $\frac{3}{10} x^{\frac{10}{3}}$ .

$\frac{3}{10} x^{\frac{10}{3}} + C + \int x^{- \frac{1}{6}} d x$

Step 7

By the Power Rule, the integral of $x^{- \frac{1}{6}}$ with respect to $x$ is $\frac{6}{5} x^{\frac{5}{6}}$ .

$\frac{3}{10} x^{\frac{10}{3}} + C + \frac{6}{5} x^{\frac{5}{6}} + C$

Step 8

Simplify.

$\frac{3}{10} x^{\frac{10}{3}} + \frac{6}{5} x^{\frac{5}{6}} + C$