Calculus Examples

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

Step 1

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

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Step 1.1

Let $u = x^{2} + 1$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x^{2} + 1$ .

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

Step 1.1.2

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

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

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [1]$

Step 1.1.4

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

$2 x + 0$

Step 1.1.5

Add $2 x$ and $0$ .

$2 x$

Step 1.2

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

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

Step 2

Simplify.

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Step 2.1

Rewrite ${\sqrt{u - 1}}^{2}$ as $u - 1$ .

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Step 2.1.1

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

$\int \frac{{({(u - 1)}^{\frac{1}{2}})}^{2}}{\sqrt[3]{u}} \cdot \frac{1}{2} d u$

Step 2.1.2

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

$\int \frac{{(u - 1)}^{\frac{1}{2} \cdot 2}}{\sqrt[3]{u}} \cdot \frac{1}{2} d u$

Step 2.1.3

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

$\int \frac{{(u - 1)}^{\frac{2}{2}}}{\sqrt[3]{u}} \cdot \frac{1}{2} d u$

Step 2.1.4

Cancel the common factor of $2$ .

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Step 2.1.4.1

Cancel the common factor.

$\int \frac{{(u - 1)}^{\frac{2}{2}}}{\sqrt[3]{u}} \cdot \frac{1}{2} d u$

Step 2.1.4.2

Rewrite the expression.

$\int \frac{{(u - 1)}^{1}}{\sqrt[3]{u}} \cdot \frac{1}{2} d u$

Step 2.1.5

Simplify.

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

Step 2.2

Multiply $\frac{u - 1}{\sqrt[3]{u}}$ by $\frac{1}{2}$ .

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

Step 2.3

Move $2$ to the left of $\sqrt[3]{u}$ .

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

Step 3

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

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

Step 4

Apply basic rules of exponents.

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Step 4.1

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

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

Step 4.2

Move $u^{\frac{1}{3}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 4.3

Multiply the exponents in ${(u^{\frac{1}{3}})}^{- 1}$ .

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

Move the negative in front of the fraction.

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

Step 5

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

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Step 5.1

Apply the distributive property.

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

Combine the numerators over the common denominator.

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

Step 5.6

Subtract $1$ from $3$ .

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

Step 6

Split the single integral into multiple integrals.

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

Step 7

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

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

Step 8

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

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

Step 9

By the Power Rule, the integral of $u^{- \frac{1}{3}}$ with respect to $u$ is $\frac{3}{2} u^{\frac{2}{3}}$ .

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

Step 10

Simplify.

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

Step 11

Replace all occurrences of $u$ with $x^{2} + 1$ .

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

Step 12

Simplify.

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Step 12.1

Simplify each term.

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Step 12.1.1

Combine $\frac{3}{5}$ and ${(x^{2} + 1)}^{\frac{5}{3}}$ .

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

Step 12.1.2

Combine ${(x^{2} + 1)}^{\frac{2}{3}}$ and $\frac{3}{2}$ .

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

Step 12.1.3

Move $3$ to the left of ${(x^{2} + 1)}^{\frac{2}{3}}$ .

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

Step 12.2

To write $\frac{3 {(x^{2} + 1)}^{\frac{5}{3}}}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 12.3

To write $- \frac{3 {(x^{2} + 1)}^{\frac{2}{3}}}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 12.4

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 12.4.1

Multiply $\frac{3 {(x^{2} + 1)}^{\frac{5}{3}}}{5}$ by $\frac{2}{2}$ .

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

Step 12.4.2

Multiply $5$ by $2$ .

$\frac{1}{2} (\frac{3 {(x^{2} + 1)}^{\frac{5}{3}} \cdot 2}{10} - \frac{3 {(x^{2} + 1)}^{\frac{2}{3}}}{2} \cdot \frac{5}{5}) + C$

Step 12.4.3

Multiply $\frac{3 {(x^{2} + 1)}^{\frac{2}{3}}}{2}$ by $\frac{5}{5}$ .

$\frac{1}{2} (\frac{3 {(x^{2} + 1)}^{\frac{5}{3}} \cdot 2}{10} - \frac{3 {(x^{2} + 1)}^{\frac{2}{3}} \cdot 5}{2 \cdot 5}) + C$

Step 12.4.4

Multiply $2$ by $5$ .

$\frac{1}{2} (\frac{3 {(x^{2} + 1)}^{\frac{5}{3}} \cdot 2}{10} - \frac{3 {(x^{2} + 1)}^{\frac{2}{3}} \cdot 5}{10}) + C$

Step 12.5

Combine the numerators over the common denominator.

$\frac{1}{2} \cdot \frac{3 {(x^{2} + 1)}^{\frac{5}{3}} \cdot 2 - 3 {(x^{2} + 1)}^{\frac{2}{3}} \cdot 5}{10} + C$

Step 12.6

Simplify the numerator.

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Step 12.6.1

Factor $3 {(x^{2} + 1)}^{\frac{2}{3}}$ out of $3 {(x^{2} + 1)}^{\frac{5}{3}} \cdot 2 - 3 {(x^{2} + 1)}^{\frac{2}{3}} \cdot 5$ .

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Step 12.6.1.1

Reorder the expression.

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Step 12.6.1.1.1

Move ${(x^{2} + 1)}^{\frac{5}{3}}$ .

$\frac{1}{2} \cdot \frac{3 \cdot 2 {(x^{2} + 1)}^{\frac{5}{3}} - 3 {(x^{2} + 1)}^{\frac{2}{3}} \cdot 5}{10} + C$

Step 12.6.1.1.2

Move ${(x^{2} + 1)}^{\frac{2}{3}}$ .

$\frac{1}{2} \cdot \frac{3 \cdot 2 {(x^{2} + 1)}^{\frac{5}{3}} - 3 \cdot 5 {(x^{2} + 1)}^{\frac{2}{3}}}{10} + C$

Step 12.6.1.2

Factor $3 {(x^{2} + 1)}^{\frac{2}{3}}$ out of $3 \cdot 2 {(x^{2} + 1)}^{\frac{5}{3}}$ .

$\frac{1}{2} \cdot \frac{3 {(x^{2} + 1)}^{\frac{2}{3}} (2 {(x^{2} + 1)}^{\frac{3}{3}}) - 3 \cdot 5 {(x^{2} + 1)}^{\frac{2}{3}}}{10} + C$

Step 12.6.1.3

Factor $3 {(x^{2} + 1)}^{\frac{2}{3}}$ out of $- 3 \cdot 5 {(x^{2} + 1)}^{\frac{2}{3}}$ .

$\frac{1}{2} \cdot \frac{3 {(x^{2} + 1)}^{\frac{2}{3}} (2 {(x^{2} + 1)}^{\frac{3}{3}}) + 3 {(x^{2} + 1)}^{\frac{2}{3}} (- 1 \cdot 5)}{10} + C$

Step 12.6.1.4

Factor $3 {(x^{2} + 1)}^{\frac{2}{3}}$ out of $3 {(x^{2} + 1)}^{\frac{2}{3}} (2 {(x^{2} + 1)}^{\frac{3}{3}}) + 3 {(x^{2} + 1)}^{\frac{2}{3}} (- 1 \cdot 5)$ .

$\frac{1}{2} \cdot \frac{3 {(x^{2} + 1)}^{\frac{2}{3}} (2 {(x^{2} + 1)}^{\frac{3}{3}} - 1 \cdot 5)}{10} + C$

Step 12.6.2

Multiply $- 1$ by $5$ .

$\frac{1}{2} \cdot \frac{3 {(x^{2} + 1)}^{\frac{2}{3}} (2 {(x^{2} + 1)}^{\frac{3}{3}} - 5)}{10} + C$

Step 12.6.3

Simplify each term.

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Step 12.6.3.1

Divide $3$ by $3$ .

$\frac{1}{2} \cdot \frac{3 {(x^{2} + 1)}^{\frac{2}{3}} (2 {(x^{2} + 1)}^{1} - 5)}{10} + C$

Step 12.6.3.2

Simplify.

$\frac{1}{2} \cdot \frac{3 {(x^{2} + 1)}^{\frac{2}{3}} (2 (x^{2} + 1) - 5)}{10} + C$

Step 12.6.3.3

Apply the distributive property.

$\frac{1}{2} \cdot \frac{3 {(x^{2} + 1)}^{\frac{2}{3}} (2 x^{2} + 2 \cdot 1 - 5)}{10} + C$

Step 12.6.3.4

Multiply $2$ by $1$ .

$\frac{1}{2} \cdot \frac{3 {(x^{2} + 1)}^{\frac{2}{3}} (2 x^{2} + 2 - 5)}{10} + C$

Step 12.6.4

Subtract $5$ from $2$ .

$\frac{1}{2} \cdot \frac{3 {(x^{2} + 1)}^{\frac{2}{3}} (2 x^{2} - 3)}{10} + C$

Step 12.7

Combine.

$\frac{1 (3 {(x^{2} + 1)}^{\frac{2}{3}} (2 x^{2} - 3))}{2 \cdot 10} + C$

Step 12.8

Multiply $3$ by $1$ .

$\frac{3 {(x^{2} + 1)}^{\frac{2}{3}} (2 x^{2} - 3)}{2 \cdot 10} + C$

Step 12.9

Multiply $2$ by $10$ .

$\frac{3 {(x^{2} + 1)}^{\frac{2}{3}} (2 x^{2} - 3)}{20} + C$