Calculus Examples

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

Step 1

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

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

Step 2

Expand ${(1 - x^{2})}^{2}$ .

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Step 2.1

Rewrite ${(1 - x^{2})}^{2}$ as $(1 - x^{2}) (1 - x^{2})$ .

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 2.4

Apply the distributive property.

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

Step 2.5

Move $x^{2}$ .

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

Step 2.6

Move $x^{2}$ .

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

Step 2.7

Multiply $1$ by $1$ .

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

Step 2.8

Multiply $- 1$ by $1$ .

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

Step 2.9

Multiply $- 1$ by $1$ .

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

Step 2.10

Multiply $- 1$ by $- 1$ .

$π \int_{0}^{1} 1 - x^{2} - x^{2} + 1 x^{2} x^{2} d x$

Step 2.11

Multiply $x^{2}$ by $1$ .

$π \int_{0}^{1} 1 - x^{2} - x^{2} + x^{2} x^{2} d x$

Step 2.12

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

$π \int_{0}^{1} 1 - x^{2} - x^{2} + x^{2 + 2} d x$

Step 2.13

Add $2$ and $2$ .

$π \int_{0}^{1} 1 - x^{2} - x^{2} + x^{4} d x$

Step 2.14

Subtract $x^{2}$ from $- x^{2}$ .

$π \int_{0}^{1} 1 - 2 x^{2} + x^{4} d x$

Step 2.15

Reorder $- 2 x^{2}$ and $x^{4}$ .

$π \int_{0}^{1} 1 + x^{4} - 2 x^{2} d x$

Step 2.16

Move $1$ .

$π \int_{0}^{1} x^{4} - 2 x^{2} + 1 d x$

Step 3

Split the single integral into multiple integrals.

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

Step 4

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

$π (\frac{1}{5} x^{5}]_{0}^{1} + \int_{0}^{1} - 2 x^{2} d x + \int_{0}^{1} d x)$

Step 5

Combine $\frac{1}{5}$ and $x^{5}$ .

$π (\frac{x^{5}}{5}]_{0}^{1} + \int_{0}^{1} - 2 x^{2} d x + \int_{0}^{1} d x)$

Step 6

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

$π (\frac{x^{5}}{5}]_{0}^{1} - 2 \int_{0}^{1} x^{2} d x + \int_{0}^{1} d x)$

Step 7

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

$π (\frac{x^{5}}{5}]_{0}^{1} - 2 (\frac{1}{3} x^{3}]_{0}^{1}) + \int_{0}^{1} d x)$

Step 8

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

$π (\frac{x^{5}}{5}]_{0}^{1} - 2 (\frac{x^{3}}{3}]_{0}^{1}) + \int_{0}^{1} d x)$

Step 9

Apply the constant rule.

$π (\frac{x^{5}}{5}]_{0}^{1} - 2 (\frac{x^{3}}{3}]_{0}^{1}) + x]_{0}^{1})$

Step 10

Simplify the answer.

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Step 10.1

Combine $\frac{x^{5}}{5}]_{0}^{1}$ and $x]_{0}^{1}$ .

$π (\frac{x^{5}}{5} + x]_{0}^{1} - 2 (\frac{x^{3}}{3}]_{0}^{1}))$

Step 10.2

Substitute and simplify.

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Step 10.2.1

Evaluate $\frac{x^{5}}{5} + x$ at $1$ and at $0$ .

$π ((\frac{1^{5}}{5} + 1) - (\frac{0^{5}}{5} + 0) - 2 (\frac{x^{3}}{3}]_{0}^{1}))$

Step 10.2.2

Evaluate $\frac{x^{3}}{3}$ at $1$ and at $0$ .

$π ((\frac{1^{5}}{5} + 1) - (\frac{0^{5}}{5} + 0) - 2 ((\frac{1^{3}}{3}) - \frac{0^{3}}{3}))$

Step 10.2.3

Simplify.

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Step 10.2.3.1

One to any power is one.

$π (\frac{1}{5} + 1 - (\frac{0^{5}}{5} + 0) - 2 ((\frac{1^{3}}{3}) - \frac{0^{3}}{3}))$

Step 10.2.3.2

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

$π (\frac{1}{5} + \frac{5}{5} - (\frac{0^{5}}{5} + 0) - 2 ((\frac{1^{3}}{3}) - \frac{0^{3}}{3}))$

Step 10.2.3.3

Combine the numerators over the common denominator.

$π (\frac{1 + 5}{5} - (\frac{0^{5}}{5} + 0) - 2 ((\frac{1^{3}}{3}) - \frac{0^{3}}{3}))$

Step 10.2.3.4

Add $1$ and $5$ .

$π (\frac{6}{5} - (\frac{0^{5}}{5} + 0) - 2 ((\frac{1^{3}}{3}) - \frac{0^{3}}{3}))$

Step 10.2.3.5

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

$π (\frac{6}{5} - (\frac{0}{5} + 0) - 2 ((\frac{1^{3}}{3}) - \frac{0^{3}}{3}))$

Step 10.2.3.6

Cancel the common factor of $0$ and $5$ .

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Step 10.2.3.6.1

Factor $5$ out of $0$ .

$π (\frac{6}{5} - (\frac{5 (0)}{5} + 0) - 2 ((\frac{1^{3}}{3}) - \frac{0^{3}}{3}))$

Step 10.2.3.6.2

Cancel the common factors.

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Step 10.2.3.6.2.1

Factor $5$ out of $5$ .

$π (\frac{6}{5} - (\frac{5 \cdot 0}{5 \cdot 1} + 0) - 2 ((\frac{1^{3}}{3}) - \frac{0^{3}}{3}))$

Step 10.2.3.6.2.2

Cancel the common factor.

$π (\frac{6}{5} - (\frac{5 \cdot 0}{5 \cdot 1} + 0) - 2 ((\frac{1^{3}}{3}) - \frac{0^{3}}{3}))$

Step 10.2.3.6.2.3

Rewrite the expression.

$π (\frac{6}{5} - (\frac{0}{1} + 0) - 2 ((\frac{1^{3}}{3}) - \frac{0^{3}}{3}))$

Step 10.2.3.6.2.4

Divide $0$ by $1$ .

$π (\frac{6}{5} - (0 + 0) - 2 ((\frac{1^{3}}{3}) - \frac{0^{3}}{3}))$

Step 10.2.3.7

Add $0$ and $0$ .

$π (\frac{6}{5} - 0 - 2 ((\frac{1^{3}}{3}) - \frac{0^{3}}{3}))$

Step 10.2.3.8

Multiply $- 1$ by $0$ .

$π (\frac{6}{5} + 0 - 2 ((\frac{1^{3}}{3}) - \frac{0^{3}}{3}))$

Step 10.2.3.9

Add $\frac{6}{5}$ and $0$ .

$π (\frac{6}{5} - 2 ((\frac{1^{3}}{3}) - \frac{0^{3}}{3}))$

Step 10.2.3.10

One to any power is one.

$π (\frac{6}{5} - 2 (\frac{1}{3} - \frac{0^{3}}{3}))$

Step 10.2.3.11

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

$π (\frac{6}{5} - 2 (\frac{1}{3} - \frac{0}{3}))$

Step 10.2.3.12

Cancel the common factor of $0$ and $3$ .

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Step 10.2.3.12.1

Factor $3$ out of $0$ .

$π (\frac{6}{5} - 2 (\frac{1}{3} - \frac{3 (0)}{3}))$

Step 10.2.3.12.2

Cancel the common factors.

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Step 10.2.3.12.2.1

Factor $3$ out of $3$ .

$π (\frac{6}{5} - 2 (\frac{1}{3} - \frac{3 \cdot 0}{3 \cdot 1}))$

Step 10.2.3.12.2.2

Cancel the common factor.

$π (\frac{6}{5} - 2 (\frac{1}{3} - \frac{3 \cdot 0}{3 \cdot 1}))$

Step 10.2.3.12.2.3

Rewrite the expression.

$π (\frac{6}{5} - 2 (\frac{1}{3} - \frac{0}{1}))$

Step 10.2.3.12.2.4

Divide $0$ by $1$ .

$π (\frac{6}{5} - 2 (\frac{1}{3} - 0))$

Step 10.2.3.13

Multiply $- 1$ by $0$ .

$π (\frac{6}{5} - 2 (\frac{1}{3} + 0))$

Step 10.2.3.14

Add $\frac{1}{3}$ and $0$ .

$π (\frac{6}{5} - 2 (\frac{1}{3}))$

Step 10.2.3.15

Combine $- 2$ and $\frac{1}{3}$ .

$π (\frac{6}{5} + \frac{- 2}{3})$

Step 10.2.3.16

Move the negative in front of the fraction.

$π (\frac{6}{5} - \frac{2}{3})$

Step 10.2.3.17

To write $\frac{6}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$π (\frac{6}{5} \cdot \frac{3}{3} - \frac{2}{3})$

Step 10.2.3.18

To write $- \frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$π (\frac{6}{5} \cdot \frac{3}{3} - \frac{2}{3} \cdot \frac{5}{5})$

Step 10.2.3.19

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.2.3.19.1

Multiply $\frac{6}{5}$ by $\frac{3}{3}$ .

$π (\frac{6 \cdot 3}{5 \cdot 3} - \frac{2}{3} \cdot \frac{5}{5})$

Step 10.2.3.19.2

Multiply $5$ by $3$ .

$π (\frac{6 \cdot 3}{15} - \frac{2}{3} \cdot \frac{5}{5})$

Step 10.2.3.19.3

Multiply $\frac{2}{3}$ by $\frac{5}{5}$ .

$π (\frac{6 \cdot 3}{15} - \frac{2 \cdot 5}{3 \cdot 5})$

Step 10.2.3.19.4

Multiply $3$ by $5$ .

$π (\frac{6 \cdot 3}{15} - \frac{2 \cdot 5}{15})$

Step 10.2.3.20

Combine the numerators over the common denominator.

$π \frac{6 \cdot 3 - 2 \cdot 5}{15}$

Step 10.2.3.21

Simplify the numerator.

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Step 10.2.3.21.1

Multiply $6$ by $3$ .

$π \frac{18 - 2 \cdot 5}{15}$

Step 10.2.3.21.2

Multiply $- 2$ by $5$ .

$π \frac{18 - 10}{15}$

Step 10.2.3.21.3

Subtract $10$ from $18$ .

$π \frac{8}{15}$

Step 10.2.3.22

Combine $π$ and $\frac{8}{15}$ .

$\frac{π \cdot 8}{15}$

Step 10.2.3.23

Move $8$ to the left of $π$ .

$\frac{8 π}{15}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{8 π}{15}$

Decimal Form:

$1.67551608 \dots$

Step 12