Calculus Examples

$\int_{0}^{π} x^{2} \cos (4 x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{2}$ and $d v = \cos (4 x)$ .

$x^{2} (\frac{1}{4} \sin (4 x))]_{0}^{π} - \int_{0}^{π} \frac{1}{4} \sin (4 x) (2 x) d x$

Step 2

Simplify.

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Step 2.1

Combine $\frac{1}{4}$ and $\sin (4 x)$ .

$x^{2} \frac{\sin (4 x)}{4}]_{0}^{π} - \int_{0}^{π} \frac{1}{4} \sin (4 x) (2 x) d x$

Step 2.2

Combine $x^{2}$ and $\frac{\sin (4 x)}{4}$ .

$\frac{x^{2} \sin (4 x)}{4}]_{0}^{π} - \int_{0}^{π} \frac{1}{4} \sin (4 x) (2 x) d x$

Step 3

Since $\frac{1}{4} \cdot 2$ is constant with respect to $x$ , move $\frac{1}{4} \cdot 2$ out of the integral.

$\frac{x^{2} \sin (4 x)}{4}]_{0}^{π} - (\frac{1}{4} \cdot 2 \int_{0}^{π} \sin (4 x) (x) d x)$

Step 4

Simplify.

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Step 4.1

Combine $\frac{1}{4}$ and $2$ .

$\frac{x^{2} \sin (4 x)}{4}]_{0}^{π} - (\frac{2}{4} \int_{0}^{π} \sin (4 x) (x) d x)$

Step 4.2

Cancel the common factor of $2$ and $4$ .

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Step 4.2.1

Factor $2$ out of $2$ .

$\frac{x^{2} \sin (4 x)}{4}]_{0}^{π} - (\frac{2 (1)}{4} \int_{0}^{π} \sin (4 x) (x) d x)$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $2$ out of $4$ .

$\frac{x^{2} \sin (4 x)}{4}]_{0}^{π} - (\frac{2 \cdot 1}{2 \cdot 2} \int_{0}^{π} \sin (4 x) (x) d x)$

Step 4.2.2.2

Cancel the common factor.

$\frac{x^{2} \sin (4 x)}{4}]_{0}^{π} - (\frac{2 \cdot 1}{2 \cdot 2} \int_{0}^{π} \sin (4 x) (x) d x)$

Step 4.2.2.3

Rewrite the expression.

$\frac{x^{2} \sin (4 x)}{4}]_{0}^{π} - (\frac{1}{2} \int_{0}^{π} \sin (4 x) (x) d x)$

$\frac{x^{2} \sin (4 x)}{4}]_{0}^{π} - \frac{1}{2} \int_{0}^{π} \sin (4 x) (x) d x$

Step 5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \sin (4 x)$ .

$\frac{x^{2} \sin (4 x)}{4}]_{0}^{π} - \frac{1}{2} (x (- \frac{1}{4} \cos (4 x))]_{0}^{π} - \int_{0}^{π} - \frac{1}{4} \cos (4 x) d x)$

Step 6

Simplify.

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Step 6.1

Combine $\cos (4 x)$ and $\frac{1}{4}$ .

$\frac{x^{2} \sin (4 x)}{4}]_{0}^{π} - \frac{1}{2} (x (- \frac{\cos (4 x)}{4})]_{0}^{π} - \int_{0}^{π} - \frac{1}{4} \cos (4 x) d x)$

Step 6.2

Combine $x$ and $\frac{\cos (4 x)}{4}$ .

$\frac{x^{2} \sin (4 x)}{4}]_{0}^{π} - \frac{1}{2} (- \frac{x \cos (4 x)}{4}]_{0}^{π} - \int_{0}^{π} - \frac{1}{4} \cos (4 x) d x)$

Step 6.3

Combine $\cos (4 x)$ and $\frac{1}{4}$ .

$\frac{x^{2} \sin (4 x)}{4}]_{0}^{π} - \frac{1}{2} (- \frac{x \cos (4 x)}{4}]_{0}^{π} - \int_{0}^{π} - \frac{\cos (4 x)}{4} d x)$

Step 7

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

$\frac{x^{2} \sin (4 x)}{4}]_{0}^{π} - \frac{1}{2} (- \frac{x \cos (4 x)}{4}]_{0}^{π} - - \int_{0}^{π} \frac{\cos (4 x)}{4} d x)$

Step 8

Simplify.

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Step 8.1

Multiply $- 1$ by $- 1$ .

$\frac{x^{2} \sin (4 x)}{4}]_{0}^{π} - \frac{1}{2} (- \frac{x \cos (4 x)}{4}]_{0}^{π} + 1 \int_{0}^{π} \frac{\cos (4 x)}{4} d x)$

Step 8.2

Multiply $\int_{0}^{π} \frac{\cos (4 x)}{4} d x$ by $1$ .

$\frac{x^{2} \sin (4 x)}{4}]_{0}^{π} - \frac{1}{2} (- \frac{x \cos (4 x)}{4}]_{0}^{π} + \int_{0}^{π} \frac{\cos (4 x)}{4} d x)$

Step 9

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

$\frac{x^{2} \sin (4 x)}{4}]_{0}^{π} - \frac{1}{2} (- \frac{x \cos (4 x)}{4}]_{0}^{π} + \frac{1}{4} \int_{0}^{π} \cos (4 x) d x)$

Step 10

Let $u = 4 x$ . Then $d u = 4 d x$ , so $\frac{1}{4} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 10.1

Let $u = 4 x$ . Find $\frac{d u}{d x}$ .

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Step 10.1.1

Differentiate $4 x$ .

$\frac{d}{d x} [4 x]$

Step 10.1.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x]$

Step 10.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$4 \cdot 1$

Step 10.1.4

Multiply $4$ by $1$ .

$4$

Step 10.2

Substitute the lower limit in for $x$ in $u = 4 x$ .

$u_{lower} = 4 \cdot 0$

Step 10.3

Multiply $4$ by $0$ .

$u_{lower} = 0$

Step 10.4

Substitute the upper limit in for $x$ in $u = 4 x$ .

$u_{upper} = 4 π$

Step 10.5

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

$u_{lower} = 0$

$u_{upper} = 4 π$

Step 10.6

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

$\frac{x^{2} \sin (4 x)}{4}]_{0}^{π} - \frac{1}{2} (- \frac{x \cos (4 x)}{4}]_{0}^{π} + \frac{1}{4} \int_{0}^{4 π} \cos (u) \frac{1}{4} d u)$

Step 11

Combine $\cos (u)$ and $\frac{1}{4}$ .

$\frac{x^{2} \sin (4 x)}{4}]_{0}^{π} - \frac{1}{2} (- \frac{x \cos (4 x)}{4}]_{0}^{π} + \frac{1}{4} \int_{0}^{4 π} \frac{\cos (u)}{4} d u)$

Step 12

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$\frac{x^{2} \sin (4 x)}{4}]_{0}^{π} - \frac{1}{2} (- \frac{x \cos (4 x)}{4}]_{0}^{π} + \frac{1}{4} (\frac{1}{4} \int_{0}^{4 π} \cos (u) d u))$

Step 13

Simplify.

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Step 13.1

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

$\frac{x^{2} \sin (4 x)}{4}]_{0}^{π} - \frac{1}{2} (- \frac{x \cos (4 x)}{4}]_{0}^{π} + \frac{1}{4 \cdot 4} \int_{0}^{4 π} \cos (u) d u)$

Step 13.2

Multiply $4$ by $4$ .

$\frac{x^{2} \sin (4 x)}{4}]_{0}^{π} - \frac{1}{2} (- \frac{x \cos (4 x)}{4}]_{0}^{π} + \frac{1}{16} \int_{0}^{4 π} \cos (u) d u)$

Step 14

The integral of $\cos (u)$ with respect to $u$ is $\sin (u)$ .

$\frac{x^{2} \sin (4 x)}{4}]_{0}^{π} - \frac{1}{2} (- \frac{x \cos (4 x)}{4}]_{0}^{π} + \frac{1}{16} \sin (u)]_{0}^{4 π})$

Step 15

Combine $\frac{1}{16}$ and $\sin (u)]_{0}^{4 π}$ .

$\frac{x^{2} \sin (4 x)}{4}]_{0}^{π} - \frac{1}{2} (- \frac{x \cos (4 x)}{4}]_{0}^{π} + \frac{\sin (u)]_{0}^{4 π}}{16})$

Step 16

Substitute and simplify.

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Step 16.1

Evaluate $\frac{x^{2} \sin (4 x)}{4}$ at $π$ and at $0$ .

$(\frac{π^{2} \sin (4 π)}{4}) - \frac{0^{2} \sin (4 \cdot 0)}{4} - \frac{1}{2} (- \frac{x \cos (4 x)}{4}]_{0}^{π} + \frac{\sin (u)]_{0}^{4 π}}{16})$

Step 16.2

Evaluate $- \frac{x \cos (4 x)}{4}$ at $π$ and at $0$ .

$(\frac{π^{2} \sin (4 π)}{4}) - \frac{0^{2} \sin (4 \cdot 0)}{4} - \frac{1}{2} ((- \frac{π \cos (4 π)}{4}) + \frac{0 \cos (4 \cdot 0)}{4} + \frac{\sin (u)]_{0}^{4 π}}{16})$

Step 16.3

Evaluate $\sin (u)$ at $4 π$ and at $0$ .

$(\frac{π^{2} \sin (4 π)}{4}) - \frac{0^{2} \sin (4 \cdot 0)}{4} - \frac{1}{2} ((- \frac{π \cos (4 π)}{4}) + \frac{0 \cos (4 \cdot 0)}{4} + \frac{(\sin (4 π)) - \sin (0)}{16})$

Step 16.4

Simplify.

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Step 16.4.1

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

$\frac{π^{2} \sin (4 π)}{4} - \frac{0 \sin (4 \cdot 0)}{4} - \frac{1}{2} ((- \frac{π \cos (4 π)}{4}) + \frac{0 \cos (4 \cdot 0)}{4} + \frac{(\sin (4 π)) - \sin (0)}{16})$

Step 16.4.2

Multiply $4$ by $0$ .

$\frac{π^{2} \sin (4 π)}{4} - \frac{0 \sin (0)}{4} - \frac{1}{2} ((- \frac{π \cos (4 π)}{4}) + \frac{0 \cos (4 \cdot 0)}{4} + \frac{(\sin (4 π)) - \sin (0)}{16})$

Step 16.4.3

Multiply $0$ by $\sin (0)$ .

$\frac{π^{2} \sin (4 π)}{4} - \frac{0}{4} - \frac{1}{2} ((- \frac{π \cos (4 π)}{4}) + \frac{0 \cos (4 \cdot 0)}{4} + \frac{(\sin (4 π)) - \sin (0)}{16})$

Step 16.4.4

Cancel the common factor of $0$ and $4$ .

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Step 16.4.4.1

Factor $4$ out of $0$ .

$\frac{π^{2} \sin (4 π)}{4} - \frac{4 (0)}{4} - \frac{1}{2} ((- \frac{π \cos (4 π)}{4}) + \frac{0 \cos (4 \cdot 0)}{4} + \frac{(\sin (4 π)) - \sin (0)}{16})$

Step 16.4.4.2

Cancel the common factors.

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Step 16.4.4.2.1

Factor $4$ out of $4$ .

$\frac{π^{2} \sin (4 π)}{4} - \frac{4 \cdot 0}{4 \cdot 1} - \frac{1}{2} ((- \frac{π \cos (4 π)}{4}) + \frac{0 \cos (4 \cdot 0)}{4} + \frac{(\sin (4 π)) - \sin (0)}{16})$

Step 16.4.4.2.2

Cancel the common factor.

$\frac{π^{2} \sin (4 π)}{4} - \frac{4 \cdot 0}{4 \cdot 1} - \frac{1}{2} ((- \frac{π \cos (4 π)}{4}) + \frac{0 \cos (4 \cdot 0)}{4} + \frac{(\sin (4 π)) - \sin (0)}{16})$

Step 16.4.4.2.3

Rewrite the expression.

$\frac{π^{2} \sin (4 π)}{4} - \frac{0}{1} - \frac{1}{2} ((- \frac{π \cos (4 π)}{4}) + \frac{0 \cos (4 \cdot 0)}{4} + \frac{(\sin (4 π)) - \sin (0)}{16})$

Step 16.4.4.2.4

Divide $0$ by $1$ .

$\frac{π^{2} \sin (4 π)}{4} - 0 - \frac{1}{2} ((- \frac{π \cos (4 π)}{4}) + \frac{0 \cos (4 \cdot 0)}{4} + \frac{(\sin (4 π)) - \sin (0)}{16})$

Step 16.4.5

Multiply $- 1$ by $0$ .

$\frac{π^{2} \sin (4 π)}{4} + 0 - \frac{1}{2} ((- \frac{π \cos (4 π)}{4}) + \frac{0 \cos (4 \cdot 0)}{4} + \frac{(\sin (4 π)) - \sin (0)}{16})$

Step 16.4.6

Add $\frac{π^{2} \sin (4 π)}{4}$ and $0$ .

$\frac{π^{2} \sin (4 π)}{4} - \frac{1}{2} ((- \frac{π \cos (4 π)}{4}) + \frac{0 \cos (4 \cdot 0)}{4} + \frac{(\sin (4 π)) - \sin (0)}{16})$

Step 16.4.7

Multiply $4$ by $0$ .

$\frac{π^{2} \sin (4 π)}{4} - \frac{1}{2} (- \frac{π \cos (4 π)}{4} + \frac{0 \cos (0)}{4} + \frac{(\sin (4 π)) - \sin (0)}{16})$

Step 16.4.8

Multiply $0$ by $\cos (0)$ .

$\frac{π^{2} \sin (4 π)}{4} - \frac{1}{2} (- \frac{π \cos (4 π)}{4} + \frac{0}{4} + \frac{(\sin (4 π)) - \sin (0)}{16})$

Step 16.4.9

Cancel the common factor of $0$ and $4$ .

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Step 16.4.9.1

Factor $4$ out of $0$ .

$\frac{π^{2} \sin (4 π)}{4} - \frac{1}{2} (- \frac{π \cos (4 π)}{4} + \frac{4 (0)}{4} + \frac{(\sin (4 π)) - \sin (0)}{16})$

Step 16.4.9.2

Cancel the common factors.

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Step 16.4.9.2.1

Factor $4$ out of $4$ .

$\frac{π^{2} \sin (4 π)}{4} - \frac{1}{2} (- \frac{π \cos (4 π)}{4} + \frac{4 \cdot 0}{4 \cdot 1} + \frac{(\sin (4 π)) - \sin (0)}{16})$

Step 16.4.9.2.2

Cancel the common factor.

$\frac{π^{2} \sin (4 π)}{4} - \frac{1}{2} (- \frac{π \cos (4 π)}{4} + \frac{4 \cdot 0}{4 \cdot 1} + \frac{(\sin (4 π)) - \sin (0)}{16})$

Step 16.4.9.2.3

Rewrite the expression.

$\frac{π^{2} \sin (4 π)}{4} - \frac{1}{2} (- \frac{π \cos (4 π)}{4} + \frac{0}{1} + \frac{(\sin (4 π)) - \sin (0)}{16})$

Step 16.4.9.2.4

Divide $0$ by $1$ .

$\frac{π^{2} \sin (4 π)}{4} - \frac{1}{2} (- \frac{π \cos (4 π)}{4} + 0 + \frac{(\sin (4 π)) - \sin (0)}{16})$

Step 16.4.10

Add $- \frac{π \cos (4 π)}{4}$ and $0$ .

$\frac{π^{2} \sin (4 π)}{4} - \frac{1}{2} (- \frac{π \cos (4 π)}{4} + \frac{(\sin (4 π)) - \sin (0)}{16})$

Step 16.4.11

To write $- \frac{π \cos (4 π)}{4}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{π^{2} \sin (4 π)}{4} - \frac{1}{2} (- \frac{π \cos (4 π)}{4} \cdot \frac{4}{4} + \frac{\sin (4 π) - \sin (0)}{16})$

Step 16.4.12

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 16.4.12.1

Multiply $\frac{π \cos (4 π)}{4}$ by $\frac{4}{4}$ .

$\frac{π^{2} \sin (4 π)}{4} - \frac{1}{2} (- \frac{π \cos (4 π) \cdot 4}{4 \cdot 4} + \frac{\sin (4 π) - \sin (0)}{16})$

Step 16.4.12.2

Multiply $4$ by $4$ .

$\frac{π^{2} \sin (4 π)}{4} - \frac{1}{2} (- \frac{π \cos (4 π) \cdot 4}{16} + \frac{\sin (4 π) - \sin (0)}{16})$

Step 16.4.13

Combine the numerators over the common denominator.

$\frac{π^{2} \sin (4 π)}{4} - \frac{1}{2} \cdot \frac{- π \cos (4 π) \cdot 4 + \sin (4 π) - \sin (0)}{16}$

Step 16.4.14

Multiply $4$ by $- 1$ .

$\frac{π^{2} \sin (4 π)}{4} - \frac{1}{2} \cdot \frac{- 4 π \cos (4 π) + \sin (4 π) - \sin (0)}{16}$

Step 16.4.15

Multiply $\frac{- 4 π \cos (4 π) + \sin (4 π) - \sin (0)}{16}$ by $\frac{1}{2}$ .

$\frac{π^{2} \sin (4 π)}{4} - \frac{- 4 π \cos (4 π) + \sin (4 π) - \sin (0)}{16 \cdot 2}$

Step 16.4.16

Multiply $16$ by $2$ .

$\frac{π^{2} \sin (4 π)}{4} - \frac{- 4 π \cos (4 π) + \sin (4 π) - \sin (0)}{32}$

Step 16.4.17

To write $\frac{π^{2} \sin (4 π)}{4}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{π^{2} \sin (4 π)}{4} \cdot \frac{8}{8} - \frac{- 4 π \cos (4 π) + \sin (4 π) - \sin (0)}{32}$

Step 16.4.18

Write each expression with a common denominator of $32$ , by multiplying each by an appropriate factor of $1$ .

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Step 16.4.18.1

Multiply $\frac{π^{2} \sin (4 π)}{4}$ by $\frac{8}{8}$ .

$\frac{π^{2} \sin (4 π) \cdot 8}{4 \cdot 8} - \frac{- 4 π \cos (4 π) + \sin (4 π) - \sin (0)}{32}$

Step 16.4.18.2

Multiply $4$ by $8$ .

$\frac{π^{2} \sin (4 π) \cdot 8}{32} - \frac{- 4 π \cos (4 π) + \sin (4 π) - \sin (0)}{32}$

Step 16.4.19

Combine the numerators over the common denominator.

$\frac{π^{2} \sin (4 π) \cdot 8 - (- 4 π \cos (4 π) + \sin (4 π) - \sin (0))}{32}$

Step 16.4.20

Move $8$ to the left of $π^{2} \sin (4 π)$ .

$\frac{8 π^{2} \sin (4 π) - (- 4 π \cos (4 π) + \sin (4 π) - \sin (0))}{32}$

Step 17

Simplify.

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Step 17.1

The exact value of $\sin (0)$ is $0$ .

$\frac{8 π^{2} \sin (4 π) - (- 4 π \cos (4 π) + \sin (4 π) - 0)}{32}$

Step 17.2

Multiply $- 1$ by $0$ .

$\frac{8 π^{2} \sin (4 π) - (- 4 π \cos (4 π) + \sin (4 π) + 0)}{32}$

Step 17.3

Add $- 4 π \cos (4 π) + \sin (4 π)$ and $0$ .

$\frac{8 π^{2} \sin (4 π) - (- 4 π \cos (4 π) + \sin (4 π))}{32}$

Step 18

Simplify.

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Step 18.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{8 π^{2} \sin (0) - (- 4 π \cos (4 π) + \sin (4 π))}{32}$

Step 18.2

The exact value of $\sin (0)$ is $0$ .

$\frac{8 π^{2} \cdot 0 - (- 4 π \cos (4 π) + \sin (4 π))}{32}$

Step 18.3

Multiply $8 π^{2} \cdot 0$ .

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Step 18.3.1

Multiply $0$ by $8$ .

$\frac{0 π^{2} - (- 4 π \cos (4 π) + \sin (4 π))}{32}$

Step 18.3.2

Multiply $0$ by $π^{2}$ .

$\frac{0 - (- 4 π \cos (4 π) + \sin (4 π))}{32}$

Step 18.4

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{0 - (- 4 π \cos (0) + \sin (4 π))}{32}$

Step 18.5

The exact value of $\cos (0)$ is $1$ .

$\frac{0 - (- 4 π \cdot 1 + \sin (4 π))}{32}$

Step 18.6

Multiply $- 4$ by $1$ .

$\frac{0 - (- 4 π + \sin (4 π))}{32}$

Step 18.7

Simplify each term.

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Step 18.7.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{0 - (- 4 π + \sin (0))}{32}$

Step 18.7.2

The exact value of $\sin (0)$ is $0$ .

$\frac{0 - (- 4 π + 0)}{32}$

Step 18.8

Add $- 4 π$ and $0$ .

$\frac{0 - (- 4 π)}{32}$

Step 18.9

Multiply $- 4$ by $- 1$ .

$\frac{0 + 4 π}{32}$

Step 18.10

Reduce the expression $\frac{0 + 4 π}{32}$ by cancelling the common factors.

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Step 18.10.1

Factor $4$ out of $0$ .

$\frac{4 (0) + 4 π}{32}$

Step 18.10.2

Factor $4$ out of $4 π$ .

$\frac{4 (0) + 4 (π)}{32}$

Step 18.10.3

Factor $4$ out of $4 (0) + 4 (π)$ .

$\frac{4 (0 + π)}{32}$

Step 18.10.4

Factor $4$ out of $32$ .

$\frac{4 (0 + π)}{4 \cdot 8}$

Step 18.10.5

Cancel the common factor.

$\frac{4 (0 + π)}{4 \cdot 8}$

Step 18.10.6

Rewrite the expression.

$\frac{0 + π}{8}$

Step 18.11

Add $0$ and $π$ .

$\frac{π}{8}$

Step 19

The result can be shown in multiple forms.

Exact Form:

$\frac{π}{8}$

Decimal Form:

$0.39269908 \dots$