Calculus Examples

$\int_{0}^{1} x^{2} {(3 x^{3} - 1)}^{4} d x$

Step 1

Let $u = 3 x^{3} - 1$ . Then $d u = 9 x^{2} d x$ , so $\frac{1}{9} d u = x^{2} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

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

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Step 1.1.1

Differentiate $3 x^{3} - 1$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

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 $n = 3$ .

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

Step 1.1.3.3

Multiply $3$ by $3$ .

$9 x^{2} + \frac{d}{d x} [- 1]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

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

$9 x^{2} + 0$

Step 1.1.4.2

Add $9 x^{2}$ and $0$ .

$9 x^{2}$

Step 1.2

Substitute the lower limit in for $x$ in $u = 3 x^{3} - 1$ .

$u_{lower} = 3 \cdot 0^{3} - 1$

Step 1.3

Simplify.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Raising $0$ to any positive power yields $0$ .

$u_{lower} = 3 \cdot 0 - 1$

Step 1.3.1.2

Multiply $3$ by $0$ .

$u_{lower} = 0 - 1$

Step 1.3.2

Subtract $1$ from $0$ .

$u_{lower} = - 1$

Step 1.4

Substitute the upper limit in for $x$ in $u = 3 x^{3} - 1$ .

$u_{upper} = 3 \cdot 1^{3} - 1$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

One to any power is one.

$u_{upper} = 3 \cdot 1 - 1$

Step 1.5.1.2

Multiply $3$ by $1$ .

$u_{upper} = 3 - 1$

Step 1.5.2

Subtract $1$ from $3$ .

$u_{upper} = 2$

Step 1.6

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

$u_{lower} = - 1$

$u_{upper} = 2$

Step 1.7

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

$\int_{- 1}^{2} u^{4} \frac{1}{9} d u$

Step 2

Combine $u^{4}$ and $\frac{1}{9}$ .

$\int_{- 1}^{2} \frac{u^{4}}{9} d u$

Step 3

Since $\frac{1}{9}$ is constant with respect to $u$ , move $\frac{1}{9}$ out of the integral.

$\frac{1}{9} \int_{- 1}^{2} u^{4} d u$

Step 4

By the Power Rule, the integral of $u^{4}$ with respect to $u$ is $\frac{1}{5} u^{5}$ .

$\frac{1}{9} \frac{1}{5} u^{5}]_{- 1}^{2}$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $\frac{1}{5} u^{5}$ at $2$ and at $- 1$ .

$\frac{1}{9} ((\frac{1}{5} \cdot 2^{5}) - \frac{1}{5} {(- 1)}^{5})$

Step 5.2

Simplify.

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Step 5.2.1

Raise $2$ to the power of $5$ .

$\frac{1}{9} (\frac{1}{5} \cdot 32 - \frac{1}{5} {(- 1)}^{5})$

Step 5.2.2

Combine $\frac{1}{5}$ and $32$ .

$\frac{1}{9} (\frac{32}{5} - \frac{1}{5} {(- 1)}^{5})$

Step 5.2.3

Raise $- 1$ to the power of $5$ .

$\frac{1}{9} (\frac{32}{5} - \frac{1}{5} \cdot - 1)$

Step 5.2.4

Multiply $- 1$ by $- 1$ .

$\frac{1}{9} (\frac{32}{5} + 1 (\frac{1}{5}))$

Step 5.2.5

Multiply $\frac{1}{5}$ by $1$ .

$\frac{1}{9} (\frac{32}{5} + \frac{1}{5})$

Step 5.2.6

Combine the numerators over the common denominator.

$\frac{1}{9} \cdot \frac{32 + 1}{5}$

Step 5.2.7

Add $32$ and $1$ .

$\frac{1}{9} \cdot \frac{33}{5}$

Step 5.2.8

Multiply $\frac{1}{9}$ by $\frac{33}{5}$ .

$\frac{33}{9 \cdot 5}$

Step 5.2.9

Multiply $9$ by $5$ .

$\frac{33}{45}$

Step 5.2.10

Cancel the common factor of $33$ and $45$ .

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Step 5.2.10.1

Factor $3$ out of $33$ .

$\frac{3 (11)}{45}$

Step 5.2.10.2

Cancel the common factors.

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Step 5.2.10.2.1

Factor $3$ out of $45$ .

$\frac{3 \cdot 11}{3 \cdot 15}$

Step 5.2.10.2.2

Cancel the common factor.

$\frac{3 \cdot 11}{3 \cdot 15}$

Step 5.2.10.2.3

Rewrite the expression.

$\frac{11}{15}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{11}{15}$

Decimal Form:

$0.7\overline{3}$

Step 7