Calculus Examples

$π \int_{0}^{1} [{(x^{4})}^{2} - {(x^{7})}^{2}] d x$

Step 1

Remove parentheses.

$π \int_{0}^{1} {(x^{4})}^{2} - {(x^{7})}^{2} d x$

Step 2

Simplify.

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Step 2.1

Multiply the exponents in ${(x^{4})}^{2}$ .

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Step 2.1.1

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

$π \int_{0}^{1} x^{4 \cdot 2} - {(x^{7})}^{2} d x$

Step 2.1.2

Multiply $4$ by $2$ .

$π \int_{0}^{1} x^{8} - {(x^{7})}^{2} d x$

Step 2.2

Multiply the exponents in ${(x^{7})}^{2}$ .

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Step 2.2.1

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

$π \int_{0}^{1} x^{8} - x^{7 \cdot 2} d x$

Step 2.2.2

Multiply $7$ by $2$ .

$π \int_{0}^{1} x^{8} - x^{14} d x$

Step 3

Split the single integral into multiple integrals.

$π (\int_{0}^{1} x^{8} d x + \int_{0}^{1} - x^{14} d x)$

Step 4

By the Power Rule, the integral of $x^{8}$ with respect to $x$ is $\frac{1}{9} x^{9}$ .

$π (\frac{1}{9} x^{9}]_{0}^{1} + \int_{0}^{1} - x^{14} d x)$

Step 5

Combine $\frac{1}{9}$ and $x^{9}$ .

$π (\frac{x^{9}}{9}]_{0}^{1} + \int_{0}^{1} - x^{14} d x)$

Step 6

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

$π (\frac{x^{9}}{9}]_{0}^{1} - \int_{0}^{1} x^{14} d x)$

Step 7

By the Power Rule, the integral of $x^{14}$ with respect to $x$ is $\frac{1}{15} x^{15}$ .

$π (\frac{x^{9}}{9}]_{0}^{1} - (\frac{1}{15} x^{15}]_{0}^{1}))$

Step 8

Simplify the answer.

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Step 8.1

Combine $\frac{1}{15}$ and $x^{15}$ .

$π (\frac{x^{9}}{9}]_{0}^{1} - (\frac{x^{15}}{15}]_{0}^{1}))$

Step 8.2

Substitute and simplify.

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Step 8.2.1

Evaluate $\frac{x^{9}}{9}$ at $1$ and at $0$ .

$π ((\frac{1^{9}}{9}) - \frac{0^{9}}{9} - (\frac{x^{15}}{15}]_{0}^{1}))$

Step 8.2.2

Evaluate $\frac{x^{15}}{15}$ at $1$ and at $0$ .

$π (\frac{1^{9}}{9} - \frac{0^{9}}{9} - (\frac{1^{15}}{15} - \frac{0^{15}}{15}))$

Step 8.2.3

Simplify.

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Step 8.2.3.1

One to any power is one.

$π (\frac{1}{9} - \frac{0^{9}}{9} - (\frac{1^{15}}{15} - \frac{0^{15}}{15}))$

Step 8.2.3.2

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

$π (\frac{1}{9} - \frac{0}{9} - (\frac{1^{15}}{15} - \frac{0^{15}}{15}))$

Step 8.2.3.3

Cancel the common factor of $0$ and $9$ .

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Step 8.2.3.3.1

Factor $9$ out of $0$ .

$π (\frac{1}{9} - \frac{9 (0)}{9} - (\frac{1^{15}}{15} - \frac{0^{15}}{15}))$

Step 8.2.3.3.2

Cancel the common factors.

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Step 8.2.3.3.2.1

Factor $9$ out of $9$ .

$π (\frac{1}{9} - \frac{9 \cdot 0}{9 \cdot 1} - (\frac{1^{15}}{15} - \frac{0^{15}}{15}))$

Step 8.2.3.3.2.2

Cancel the common factor.

$π (\frac{1}{9} - \frac{9 \cdot 0}{9 \cdot 1} - (\frac{1^{15}}{15} - \frac{0^{15}}{15}))$

Step 8.2.3.3.2.3

Rewrite the expression.

$π (\frac{1}{9} - \frac{0}{1} - (\frac{1^{15}}{15} - \frac{0^{15}}{15}))$

Step 8.2.3.3.2.4

Divide $0$ by $1$ .

$π (\frac{1}{9} - 0 - (\frac{1^{15}}{15} - \frac{0^{15}}{15}))$

Step 8.2.3.4

Multiply $- 1$ by $0$ .

$π (\frac{1}{9} + 0 - (\frac{1^{15}}{15} - \frac{0^{15}}{15}))$

Step 8.2.3.5

Add $\frac{1}{9}$ and $0$ .

$π (\frac{1}{9} - (\frac{1^{15}}{15} - \frac{0^{15}}{15}))$

Step 8.2.3.6

One to any power is one.

$π (\frac{1}{9} - (\frac{1}{15} - \frac{0^{15}}{15}))$

Step 8.2.3.7

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

$π (\frac{1}{9} - (\frac{1}{15} - \frac{0}{15}))$

Step 8.2.3.8

Cancel the common factor of $0$ and $15$ .

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Step 8.2.3.8.1

Factor $15$ out of $0$ .

$π (\frac{1}{9} - (\frac{1}{15} - \frac{15 (0)}{15}))$

Step 8.2.3.8.2

Cancel the common factors.

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Step 8.2.3.8.2.1

Factor $15$ out of $15$ .

$π (\frac{1}{9} - (\frac{1}{15} - \frac{15 \cdot 0}{15 \cdot 1}))$

Step 8.2.3.8.2.2

Cancel the common factor.

$π (\frac{1}{9} - (\frac{1}{15} - \frac{15 \cdot 0}{15 \cdot 1}))$

Step 8.2.3.8.2.3

Rewrite the expression.

$π (\frac{1}{9} - (\frac{1}{15} - \frac{0}{1}))$

Step 8.2.3.8.2.4

Divide $0$ by $1$ .

$π (\frac{1}{9} - (\frac{1}{15} - 0))$

Step 8.2.3.9

Multiply $- 1$ by $0$ .

$π (\frac{1}{9} - (\frac{1}{15} + 0))$

Step 8.2.3.10

Add $\frac{1}{15}$ and $0$ .

$π (\frac{1}{9} - \frac{1}{15})$

Step 8.2.3.11

To write $\frac{1}{9}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$π (\frac{1}{9} \cdot \frac{5}{5} - \frac{1}{15})$

Step 8.2.3.12

To write $- \frac{1}{15}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$π (\frac{1}{9} \cdot \frac{5}{5} - \frac{1}{15} \cdot \frac{3}{3})$

Step 8.2.3.13

Write each expression with a common denominator of $45$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.2.3.13.1

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

$π (\frac{5}{9 \cdot 5} - \frac{1}{15} \cdot \frac{3}{3})$

Step 8.2.3.13.2

Multiply $9$ by $5$ .

$π (\frac{5}{45} - \frac{1}{15} \cdot \frac{3}{3})$

Step 8.2.3.13.3

Multiply $\frac{1}{15}$ by $\frac{3}{3}$ .

$π (\frac{5}{45} - \frac{3}{15 \cdot 3})$

Step 8.2.3.13.4

Multiply $15$ by $3$ .

$π (\frac{5}{45} - \frac{3}{45})$

Step 8.2.3.14

Combine the numerators over the common denominator.

$π \frac{5 - 3}{45}$

Step 8.2.3.15

Subtract $3$ from $5$ .

$π \frac{2}{45}$

Step 8.2.3.16

Combine $π$ and $\frac{2}{45}$ .

$\frac{π \cdot 2}{45}$

Step 8.2.3.17

Move $2$ to the left of $π$ .

$\frac{2 π}{45}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{2 π}{45}$

Decimal Form:

$0.13962634 \dots$

Step 10