Calculus Examples

$\int x^{3} \sqrt[3]{x^{2} + 4} d x$

Step 1

Let $u = x^{2} + 4$ . 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 $u = x^{2} + 4$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x^{2} + 4$ .

$\frac{d}{d x} [x^{2} + 4]$

Step 1.1.2

By the Sum Rule, the derivative of $x^{2} + 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4]$

Step 1.1.3

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

$2 x + \frac{d}{d x} [4]$

Step 1.1.4

Since $4$ is constant with respect to $x$ , the derivative of $4$ 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 $d u$ .

$\int {\sqrt{u - 4}}^{2} \sqrt[3]{u} \frac{1}{2} d u$

Step 2

Simplify the expression.

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Step 2.1

Simplify.

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Step 2.1.1

Rewrite ${\sqrt{u - 4}}^{2}$ as $u - 4$ .

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Step 2.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u - 4}$ as ${(u - 4)}^{\frac{1}{2}}$ .

$\int {({(u - 4)}^{\frac{1}{2}})}^{2} \sqrt[3]{u} \frac{1}{2} d u$

Step 2.1.1.2

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

$\int {(u - 4)}^{\frac{1}{2} \cdot 2} \sqrt[3]{u} \frac{1}{2} d u$

Step 2.1.1.3

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

$\int {(u - 4)}^{\frac{2}{2}} \sqrt[3]{u} \frac{1}{2} d u$

Step 2.1.1.4

Cancel the common factor of $2$ .

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Step 2.1.1.4.1

Cancel the common factor.

$\int {(u - 4)}^{\frac{2}{2}} \sqrt[3]{u} \frac{1}{2} d u$

Step 2.1.1.4.2

Rewrite the expression.

$\int {(u - 4)}^{1} \sqrt[3]{u} \frac{1}{2} d u$

Step 2.1.1.5

Simplify.

$\int (u - 4) \sqrt[3]{u} \frac{1}{2} d u$

Step 2.1.2

Combine $\frac{1}{2}$ and $\sqrt[3]{u}$ .

$\int (u - 4) \frac{\sqrt[3]{u}}{2} d u$

Step 2.2

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

$\int (u - 4) \frac{u^{\frac{1}{3}}}{2} d u$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\int u \frac{u^{\frac{1}{3}}}{2} - 4 \frac{u^{\frac{1}{3}}}{2} d u$

Step 3.2

Combine $u$ and $\frac{u^{\frac{1}{3}}}{2}$ .

$\int \frac{u \cdot u^{\frac{1}{3}}}{2} - 4 \frac{u^{\frac{1}{3}}}{2} d u$

Step 3.3

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

$\int \frac{u^{1} u^{\frac{1}{3}}}{2} - 4 \frac{u^{\frac{1}{3}}}{2} d u$

Step 3.4

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

$\int \frac{u^{1 + \frac{1}{3}}}{2} - 4 \frac{u^{\frac{1}{3}}}{2} d u$

Step 3.5

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

$\int \frac{u^{\frac{3}{3} + \frac{1}{3}}}{2} - 4 \frac{u^{\frac{1}{3}}}{2} d u$

Step 3.6

Combine the numerators over the common denominator.

$\int \frac{u^{\frac{3 + 1}{3}}}{2} - 4 \frac{u^{\frac{1}{3}}}{2} d u$

Step 3.7

Add $3$ and $1$ .

$\int \frac{u^{\frac{4}{3}}}{2} - 4 \frac{u^{\frac{1}{3}}}{2} d u$

Step 3.8

Combine $- 4$ and $\frac{u^{\frac{1}{3}}}{2}$ .

$\int \frac{u^{\frac{4}{3}}}{2} + \frac{- 4 u^{\frac{1}{3}}}{2} d u$

Step 4

Simplify.

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Step 4.1

Factor $2$ out of $- 4 u^{\frac{1}{3}}$ .

$\int \frac{u^{\frac{4}{3}}}{2} + \frac{2 (- 2 u^{\frac{1}{3}})}{2} d u$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $2$ out of $2$ .

$\int \frac{u^{\frac{4}{3}}}{2} + \frac{2 (- 2 u^{\frac{1}{3}})}{2 (1)} d u$

Step 4.2.2

Cancel the common factor.

$\int \frac{u^{\frac{4}{3}}}{2} + \frac{2 (- 2 u^{\frac{1}{3}})}{2 \cdot 1} d u$

Step 4.2.3

Rewrite the expression.

$\int \frac{u^{\frac{4}{3}}}{2} + \frac{- 2 u^{\frac{1}{3}}}{1} d u$

Step 4.2.4

Divide $- 2 u^{\frac{1}{3}}$ by $1$ .

$\int \frac{u^{\frac{4}{3}}}{2} - 2 u^{\frac{1}{3}} d u$

Step 5

Split the single integral into multiple integrals.

$\int \frac{u^{\frac{4}{3}}}{2} d u + \int - 2 u^{\frac{1}{3}} d u$

Step 6

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

$\frac{1}{2} \int u^{\frac{4}{3}} d u + \int - 2 u^{\frac{1}{3}} d u$

Step 7

By the Power Rule, the integral of $u^{\frac{4}{3}}$ with respect to $u$ is $\frac{3}{7} u^{\frac{7}{3}}$ .

$\frac{1}{2} (\frac{3}{7} u^{\frac{7}{3}} + C) + \int - 2 u^{\frac{1}{3}} d u$

Step 8

Since $- 2$ is constant with respect to $u$ , move $- 2$ out of the integral.

$\frac{1}{2} (\frac{3}{7} u^{\frac{7}{3}} + C) - 2 \int u^{\frac{1}{3}} d u$

Step 9

By the Power Rule, the integral of $u^{\frac{1}{3}}$ with respect to $u$ is $\frac{3}{4} u^{\frac{4}{3}}$ .

$\frac{1}{2} (\frac{3}{7} u^{\frac{7}{3}} + C) - 2 (\frac{3}{4} u^{\frac{4}{3}} + C)$

Step 10

Simplify.

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Step 10.1

Simplify.

$\frac{1}{2} \cdot \frac{3}{7} u^{\frac{7}{3}} - 2 (\frac{3}{4}) u^{\frac{4}{3}} + C$

Step 10.2

Rewrite $\frac{1}{2} \cdot \frac{3}{7} u^{\frac{7}{3}} - 2 (\frac{3}{4}) u^{\frac{4}{3}} + C$ as $\frac{3}{14} u^{\frac{7}{3}} - 2 (\frac{3}{4}) u^{\frac{4}{3}} + C$ .

$\frac{3}{14} u^{\frac{7}{3}} - 2 (\frac{3}{4}) u^{\frac{4}{3}} + C$

Step 10.3

Simplify.

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Step 10.3.1

Combine $- 2$ and $\frac{3}{4}$ .

$\frac{3}{14} u^{\frac{7}{3}} + \frac{- 2 \cdot 3}{4} u^{\frac{4}{3}} + C$

Step 10.3.2

Multiply $- 2$ by $3$ .

$\frac{3}{14} u^{\frac{7}{3}} + \frac{- 6}{4} u^{\frac{4}{3}} + C$

Step 10.3.3

Cancel the common factor of $- 6$ and $4$ .

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Step 10.3.3.1

Factor $2$ out of $- 6$ .

$\frac{3}{14} u^{\frac{7}{3}} + \frac{2 (- 3)}{4} u^{\frac{4}{3}} + C$

Step 10.3.3.2

Cancel the common factors.

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Step 10.3.3.2.1

Factor $2$ out of $4$ .

$\frac{3}{14} u^{\frac{7}{3}} + \frac{2 \cdot - 3}{2 \cdot 2} u^{\frac{4}{3}} + C$

Step 10.3.3.2.2

Cancel the common factor.

$\frac{3}{14} u^{\frac{7}{3}} + \frac{2 \cdot - 3}{2 \cdot 2} u^{\frac{4}{3}} + C$

Step 10.3.3.2.3

Rewrite the expression.

$\frac{3}{14} u^{\frac{7}{3}} + \frac{- 3}{2} u^{\frac{4}{3}} + C$

Step 10.3.4

Move the negative in front of the fraction.

$\frac{3}{14} u^{\frac{7}{3}} - \frac{3}{2} u^{\frac{4}{3}} + C$

Step 11

Replace all occurrences of $u$ with $x^{2} + 4$ .

$\frac{3}{14} {(x^{2} + 4)}^{\frac{7}{3}} - \frac{3}{2} {(x^{2} + 4)}^{\frac{4}{3}} + C$