Calculus Examples

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

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 $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Move the negative in front of the fraction.

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

Step 2

Let $u_{1} = 1 + 3 x$ . Then $d u_{1} = 3 d x$ , so $\frac{1}{3} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 2.1

Let $u_{1} = 1 + 3 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 2.1.1

Differentiate $1 + 3 x$ .

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

Step 2.1.2

Differentiate.

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Step 2.1.2.1

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

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

Step 2.1.2.2

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

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

Step 2.1.3

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

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Step 2.1.3.1

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

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

Step 2.1.3.2

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

$0 + 3 \cdot 1$

Step 2.1.3.3

Multiply $3$ by $1$ .

$0 + 3$

Step 2.1.4

Add $0$ and $3$ .

$3$

Step 2.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

Combine $\frac{{u_{1}}^{2}}{3}$ and ${(\frac{u_{1}}{3} - \frac{1}{3})}^{- \frac{1}{3}}$ .

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

Step 3.3

Move ${(\frac{u_{1}}{3} - \frac{1}{3})}^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4

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

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

Step 5

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

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Step 5.1

Let $u_{2} = \frac{u_{1}}{3} - \frac{1}{3}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 5.1.1

Differentiate $\frac{u_{1}}{3} - \frac{1}{3}$ .

$\frac{d}{d u_{1}} [\frac{u_{1}}{3} - \frac{1}{3}]$

Step 5.1.2

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

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

Step 5.1.3

Evaluate $\frac{d}{d u_{1}} [\frac{u_{1}}{3}]$ .

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Step 5.1.3.1

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

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

Step 5.1.3.2

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

$\frac{1}{3} \cdot 1 + \frac{d}{d u_{1}} [- \frac{1}{3}]$

Step 5.1.3.3

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

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

Step 5.1.4

Differentiate using the Constant Rule.

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Step 5.1.4.1

Since $- \frac{1}{3}$ is constant with respect to $u_{1}$ , the derivative of $- \frac{1}{3}$ with respect to $u_{1}$ is $0$ .

$\frac{1}{3} + 0$

Step 5.1.4.2

Add $\frac{1}{3}$ and $0$ .

$\frac{1}{3}$

Step 5.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

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

Step 6

Simplify.

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Step 6.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{3}$ .

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

Step 6.2

Multiply $3$ by $1$ .

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

Step 6.3

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

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

Step 6.4

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

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

Step 7

Since $3$ is constant with respect to $u_{2}$ , move $3$ out of the integral.

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

Step 8

Simplify the expression.

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Step 8.1

Simplify.

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Step 8.1.1

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

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

Step 8.1.2

Cancel the common factor of $3$ .

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Step 8.1.2.1

Cancel the common factor.

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

Step 8.1.2.2

Rewrite the expression.

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

Step 8.1.3

Multiply $\int \frac{{(3 u_{2} + 1)}^{2}}{{u_{2}}^{\frac{1}{3}}} d u_{2}$ by $1$ .

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

Step 8.2

Apply basic rules of exponents.

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Step 8.2.1

Move ${u_{2}}^{\frac{1}{3}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 8.2.2

Multiply the exponents in ${({u_{2}}^{\frac{1}{3}})}^{- 1}$ .

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Step 8.2.2.1

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

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

Step 8.2.2.2

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

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

Step 8.2.2.3

Move the negative in front of the fraction.

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

Step 9

Simplify.

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Step 9.1

Rewrite ${(3 u_{2} + 1)}^{2}$ as $(3 u_{2} + 1) (3 u_{2} + 1)$ .

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

Step 9.2