Calculus Examples

$\int_{0}^{1} 14 \sqrt[3]{1 + 7 x} d x$

Step 1

Since $14$ is constant with respect to $x$ , move $14$ out of the integral.

$14 \int_{0}^{1} \sqrt[3]{1 + 7 x} d x$

Step 2

Let $u = 1 + 7 x$ . Then $d u = 7 d x$ , so $\frac{1}{7} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 1 + 7 x$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $1 + 7 x$ .

$\frac{d}{d x} [1 + 7 x]$

Step 2.1.2

Differentiate.

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Step 2.1.2.1

By the Sum Rule, the derivative of $1 + 7 x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [7 x]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [7 x]$

Step 2.1.2.2

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

$0 + \frac{d}{d x} [7 x]$

Step 2.1.3

Evaluate $\frac{d}{d x} [7 x]$ .

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Step 2.1.3.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$0 + 7 \frac{d}{d x} [x]$

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 $n = 1$ .

$0 + 7 \cdot 1$

Step 2.1.3.3

Multiply $7$ by $1$ .

$0 + 7$

Step 2.1.4

Add $0$ and $7$ .

$7$

Step 2.2

Substitute the lower limit in for $x$ in $u = 1 + 7 x$ .

$u_{lower} = 1 + 7 \cdot 0$

Step 2.3

Simplify.

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Step 2.3.1

Multiply $7$ by $0$ .

$u_{lower} = 1 + 0$

Step 2.3.2

Add $1$ and $0$ .

$u_{lower} = 1$

Step 2.4

Substitute the upper limit in for $x$ in $u = 1 + 7 x$ .

$u_{upper} = 1 + 7 \cdot 1$

Step 2.5

Simplify.

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Step 2.5.1

Multiply $7$ by $1$ .

$u_{upper} = 1 + 7$

Step 2.5.2

Add $1$ and $7$ .

$u_{upper} = 8$

Step 2.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 1$

$u_{upper} = 8$

Step 2.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$14 \int_{1}^{8} \sqrt[3]{u} \frac{1}{7} d u$

Step 3

Combine $\sqrt[3]{u}$ and $\frac{1}{7}$ .

$14 \int_{1}^{8} \frac{\sqrt[3]{u}}{7} d u$

Step 4

Since $\frac{1}{7}$ is constant with respect to $u$ , move $\frac{1}{7}$ out of the integral.

$14 (\frac{1}{7} \int_{1}^{8} \sqrt[3]{u} d u)$

Step 5

Simplify the expression.

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Step 5.1

Simplify.

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Step 5.1.1

Combine $\frac{1}{7}$ and $14$ .

$\frac{14}{7} \int_{1}^{8} \sqrt[3]{u} d u$

Step 5.1.2

Cancel the common factor of $14$ and $7$ .

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Step 5.1.2.1

Factor $7$ out of $14$ .

$\frac{7 \cdot 2}{7} \int_{1}^{8} \sqrt[3]{u} d u$

Step 5.1.2.2

Cancel the common factors.

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Step 5.1.2.2.1

Factor $7$ out of $7$ .

$\frac{7 \cdot 2}{7 (1)} \int_{1}^{8} \sqrt[3]{u} d u$

Step 5.1.2.2.2

Cancel the common factor.

$\frac{7 \cdot 2}{7 \cdot 1} \int_{1}^{8} \sqrt[3]{u} d u$

Step 5.1.2.2.3

Rewrite the expression.

$\frac{2}{1} \int_{1}^{8} \sqrt[3]{u} d u$

Step 5.1.2.2.4

Divide $2$ by $1$ .

$2 \int_{1}^{8} \sqrt[3]{u} d u$

Step 5.2

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

$2 \int_{1}^{8} u^{\frac{1}{3}} d u$

Step 6

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

$2 (\frac{3}{4} u^{\frac{4}{3}}]_{1}^{8})$

Step 7

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

$2 (\frac{3 u^{\frac{4}{3}}}{4}]_{1}^{8})$

Step 8

Substitute and simplify.

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Step 8.1

Evaluate $\frac{3 u^{\frac{4}{3}}}{4}$ at $8$ and at $1$ .

$2 ((\frac{3 \cdot 8^{\frac{4}{3}}}{4}) - \frac{3 \cdot 1^{\frac{4}{3}}}{4})$

Step 8.2

Simplify.

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Step 8.2.1

Rewrite $8$ as $2^{3}$ .

$2 (\frac{3 \cdot {(2^{3})}^{\frac{4}{3}}}{4} - \frac{3 \cdot 1^{\frac{4}{3}}}{4})$

Step 8.2.2

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

$2 (\frac{3 \cdot 2^{3 (\frac{4}{3})}}{4} - \frac{3 \cdot 1^{\frac{4}{3}}}{4})$

Step 8.2.3

Cancel the common factor of $3$ .

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Step 8.2.3.1

Cancel the common factor.

$2 (\frac{3 \cdot 2^{3 (\frac{4}{3})}}{4} - \frac{3 \cdot 1^{\frac{4}{3}}}{4})$

Step 8.2.3.2

Rewrite the expression.

$2 (\frac{3 \cdot 2^{4}}{4} - \frac{3 \cdot 1^{\frac{4}{3}}}{4})$

Step 8.2.4

Raise $2$ to the power of $4$ .

$2 (\frac{3 \cdot 16}{4} - \frac{3 \cdot 1^{\frac{4}{3}}}{4})$

Step 8.2.5

Multiply $3$ by $16$ .

$2 (\frac{48}{4} - \frac{3 \cdot 1^{\frac{4}{3}}}{4})$

Step 8.2.6

Cancel the common factor of $48$ and $4$ .

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Step 8.2.6.1

Factor $4$ out of $48$ .

$2 (\frac{4 \cdot 12}{4} - \frac{3 \cdot 1^{\frac{4}{3}}}{4})$

Step 8.2.6.2

Cancel the common factors.

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Step 8.2.6.2.1

Factor $4$ out of $4$ .

$2 (\frac{4 \cdot 12}{4 (1)} - \frac{3 \cdot 1^{\frac{4}{3}}}{4})$

Step 8.2.6.2.2

Cancel the common factor.

$2 (\frac{4 \cdot 12}{4 \cdot 1} - \frac{3 \cdot 1^{\frac{4}{3}}}{4})$

Step 8.2.6.2.3

Rewrite the expression.

$2 (\frac{12}{1} - \frac{3 \cdot 1^{\frac{4}{3}}}{4})$

Step 8.2.6.2.4

Divide $12$ by $1$ .

$2 (12 - \frac{3 \cdot 1^{\frac{4}{3}}}{4})$

Step 8.2.7

One to any power is one.

$2 (12 - \frac{3 \cdot 1}{4})$

Step 8.2.8

Multiply $3$ by $1$ .

$2 (12 - \frac{3}{4})$

Step 8.2.9

To write $12$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$2 (12 \cdot \frac{4}{4} - \frac{3}{4})$

Step 8.2.10

Combine $12$ and $\frac{4}{4}$ .

$2 (\frac{12 \cdot 4}{4} - \frac{3}{4})$

Step 8.2.11

Combine the numerators over the common denominator.

$2 \frac{12 \cdot 4 - 3}{4}$

Step 8.2.12

Simplify the numerator.

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Step 8.2.12.1

Multiply $12$ by $4$ .

$2 \frac{48 - 3}{4}$

Step 8.2.12.2

Subtract $3$ from $48$ .

$2 (\frac{45}{4})$

Step 8.2.13

Combine $2$ and $\frac{45}{4}$ .

$\frac{2 \cdot 45}{4}$

Step 8.2.14

Multiply $2$ by $45$ .

$\frac{90}{4}$

Step 8.2.15

Cancel the common factor of $90$ and $4$ .

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Step 8.2.15.1

Factor $2$ out of $90$ .

$\frac{2 (45)}{4}$

Step 8.2.15.2

Cancel the common factors.

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Step 8.2.15.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 45}{2 \cdot 2}$

Step 8.2.15.2.2

Cancel the common factor.

$\frac{2 \cdot 45}{2 \cdot 2}$

Step 8.2.15.2.3

Rewrite the expression.

$\frac{45}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{45}{2}$

Decimal Form:

$22.5$

Mixed Number Form:

$22 \frac{1}{2}$

Step 10