Calculus Examples

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

Step 1

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 1.1

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

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Step 1.1.1

Differentiate $1 + 7 x$ .

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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 1.1.2.2

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 1.1.3

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

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Step 1.1.3.1

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 1.1.3.2

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 1.1.3.3

Multiply $7$ by $1$ .

$0 + 7$

Step 1.1.4

Add $0$ and $7$ .

$7$

Step 1.2

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

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

Step 1.3

Simplify.

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Step 1.3.1

Multiply $7$ by $0$ .

$u_{lower} = 1 + 0$

Step 1.3.2

Add $1$ and $0$ .

$u_{lower} = 1$

Step 1.4

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

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

Step 1.5

Simplify.

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Step 1.5.1

Multiply $7$ by $1$ .

$u_{upper} = 1 + 7$

Step 1.5.2

Add $1$ and $7$ .

$u_{upper} = 8$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = 8$

Step 1.7

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

$\frac{1}{7} \int_{1}^{8} u^{\frac{1}{3}} d u$

Step 5

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

$\frac{1}{7} \frac{3}{4} u^{\frac{4}{3}}]_{1}^{8}$

Step 6

Substitute and simplify.

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Step 6.1

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

$\frac{1}{7} ((\frac{3}{4} \cdot 8^{\frac{4}{3}}) - \frac{3}{4} \cdot 1^{\frac{4}{3}})$

Step 6.2

Simplify.

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Step 6.2.1

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

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

Step 6.2.2

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

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

Step 6.2.3

Cancel the common factor of $3$ .

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Step 6.2.3.1

Cancel the common factor.

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

Step 6.2.3.2

Rewrite the expression.

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

Step 6.2.4

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

$\frac{1}{7} (\frac{3}{4} \cdot 16 - \frac{3}{4} \cdot 1^{\frac{4}{3}})$

Step 6.2.5

Combine $\frac{3}{4}$ and $16$ .

$\frac{1}{7} (\frac{3 \cdot 16}{4} - \frac{3}{4} \cdot 1^{\frac{4}{3}})$

Step 6.2.6

Multiply $3$ by $16$ .

$\frac{1}{7} (\frac{48}{4} - \frac{3}{4} \cdot 1^{\frac{4}{3}})$

Step 6.2.7

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

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Step 6.2.7.1

Factor $4$ out of $48$ .

$\frac{1}{7} (\frac{4 \cdot 12}{4} - \frac{3}{4} \cdot 1^{\frac{4}{3}})$

Step 6.2.7.2

Cancel the common factors.

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Step 6.2.7.2.1

Factor $4$ out of $4$ .

$\frac{1}{7} (\frac{4 \cdot 12}{4 (1)} - \frac{3}{4} \cdot 1^{\frac{4}{3}})$

Step 6.2.7.2.2

Cancel the common factor.

$\frac{1}{7} (\frac{4 \cdot 12}{4 \cdot 1} - \frac{3}{4} \cdot 1^{\frac{4}{3}})$

Step 6.2.7.2.3

Rewrite the expression.

$\frac{1}{7} (\frac{12}{1} - \frac{3}{4} \cdot 1^{\frac{4}{3}})$

Step 6.2.7.2.4

Divide $12$ by $1$ .

$\frac{1}{7} (12 - \frac{3}{4} \cdot 1^{\frac{4}{3}})$

Step 6.2.8

One to any power is one.

$\frac{1}{7} (12 - \frac{3}{4} \cdot 1)$

Step 6.2.9

Multiply $- 1$ by $1$ .

$\frac{1}{7} (12 - \frac{3}{4})$

Step 6.2.10

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

$\frac{1}{7} (12 \cdot \frac{4}{4} - \frac{3}{4})$

Step 6.2.11

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

$\frac{1}{7} (\frac{12 \cdot 4}{4} - \frac{3}{4})$

Step 6.2.12

Combine the numerators over the common denominator.

$\frac{1}{7} \cdot \frac{12 \cdot 4 - 3}{4}$

Step 6.2.13

Simplify the numerator.

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Step 6.2.13.1

Multiply $12$ by $4$ .

$\frac{1}{7} \cdot \frac{48 - 3}{4}$

Step 6.2.13.2

Subtract $3$ from $48$ .

$\frac{1}{7} \cdot \frac{45}{4}$

Step 6.2.14

Multiply $\frac{1}{7}$ by $\frac{45}{4}$ .

$\frac{45}{7 \cdot 4}$

Step 6.2.15

Multiply $7$ by $4$ .

$\frac{45}{28}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{45}{28}$

Decimal Form:

$1.60714285 \dots$

Mixed Number Form:

$1 \frac{17}{28}$

Step 8