Calculus Examples

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

Step 1

Let $u = x^{3} + 1$ . Then $d u = 3 x^{2} d x$ , so $\frac{1}{3} d u = x^{2} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

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

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Step 1.1.1

Differentiate $x^{3} + 1$ .

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

Step 1.1.2

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

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

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 1.1.4

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

$3 x^{2} + 0$

Step 1.1.5

Add $3 x^{2}$ and $0$ .

$3 x^{2}$

Step 1.2

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

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

Step 2

Simplify.

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Step 2.1

Rewrite ${\sqrt[3]{u - 1}}^{3}$ as $u - 1$ .

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

Cancel the common factor of $3$ .

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Step 2.1.4.1

Cancel the common factor.

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

Step 2.1.4.2

Rewrite the expression.

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

Step 2.1.5

Simplify.

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

Step 2.2

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

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Combine the numerators over the common denominator.

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

Step 3.7

Add $2$ and $1$ .

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

Step 4

Rewrite $- 1 \frac{u^{\frac{1}{2}}}{3}$ as $- \frac{u^{\frac{1}{2}}}{3}$ .

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify.

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Step 11.1

Simplify.

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

Step 11.2

Rewrite $\frac{1}{3} \cdot \frac{2}{5} u^{\frac{5}{2}} - \frac{1}{3} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$ as $\frac{2}{15} u^{\frac{5}{2}} - \frac{1}{3} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$ .

$\frac{2}{15} u^{\frac{5}{2}} - \frac{1}{3} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$

Step 11.3

Simplify.

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Step 11.3.1

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

$\frac{2}{15} u^{\frac{5}{2}} - \frac{2}{3 \cdot 3} u^{\frac{3}{2}} + C$

Step 11.3.2

Multiply $3$ by $3$ .

$\frac{2}{15} u^{\frac{5}{2}} - \frac{2}{9} u^{\frac{3}{2}} + C$

Step 12

Replace all occurrences of $u$ with $x^{3} + 1$ .

$\frac{2}{15} {(x^{3} + 1)}^{\frac{5}{2}} - \frac{2}{9} {(x^{3} + 1)}^{\frac{3}{2}} + C$