Calculus Examples

$\int_{0}^{\frac{π}{6}} x^{2} \cos (3 x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{2}$ and $d v = \cos (3 x)$ .

$x^{2} (\frac{1}{3} \sin (3 x))]_{0}^{\frac{π}{6}} - \int_{0}^{\frac{π}{6}} \frac{1}{3} \sin (3 x) (2 x) d x$

Step 2

Simplify.

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Step 2.1

Combine $\frac{1}{3}$ and $\sin (3 x)$ .

$x^{2} \frac{\sin (3 x)}{3}]_{0}^{\frac{π}{6}} - \int_{0}^{\frac{π}{6}} \frac{1}{3} \sin (3 x) (2 x) d x$

Step 2.2

Combine $x^{2}$ and $\frac{\sin (3 x)}{3}$ .

$\frac{x^{2} \sin (3 x)}{3}]_{0}^{\frac{π}{6}} - \int_{0}^{\frac{π}{6}} \frac{1}{3} \sin (3 x) (2 x) d x$

Step 3

Since $\frac{1}{3} \cdot 2$ is constant with respect to $x$ , move $\frac{1}{3} \cdot 2$ out of the integral.

$\frac{x^{2} \sin (3 x)}{3}]_{0}^{\frac{π}{6}} - (\frac{1}{3} \cdot 2 \int_{0}^{\frac{π}{6}} \sin (3 x) (x) d x)$

Step 4

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

$\frac{x^{2} \sin (3 x)}{3}]_{0}^{\frac{π}{6}} - \frac{2}{3} \int_{0}^{\frac{π}{6}} \sin (3 x) (x) d x$

Step 5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \sin (3 x)$ .

$\frac{x^{2} \sin (3 x)}{3}]_{0}^{\frac{π}{6}} - \frac{2}{3} (x (- \frac{1}{3} \cos (3 x))]_{0}^{\frac{π}{6}} - \int_{0}^{\frac{π}{6}} - \frac{1}{3} \cos (3 x) d x)$

Step 6

Simplify.

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Step 6.1

Combine $\cos (3 x)$ and $\frac{1}{3}$ .

$\frac{x^{2} \sin (3 x)}{3}]_{0}^{\frac{π}{6}} - \frac{2}{3} (x (- \frac{\cos (3 x)}{3})]_{0}^{\frac{π}{6}} - \int_{0}^{\frac{π}{6}} - \frac{1}{3} \cos (3 x) d x)$

Step 6.2

Combine $x$ and $\frac{\cos (3 x)}{3}$ .

$\frac{x^{2} \sin (3 x)}{3}]_{0}^{\frac{π}{6}} - \frac{2}{3} (- \frac{x \cos (3 x)}{3}]_{0}^{\frac{π}{6}} - \int_{0}^{\frac{π}{6}} - \frac{1}{3} \cos (3 x) d x)$

Step 6.3

Combine $\cos (3 x)$ and $\frac{1}{3}$ .

$\frac{x^{2} \sin (3 x)}{3}]_{0}^{\frac{π}{6}} - \frac{2}{3} (- \frac{x \cos (3 x)}{3}]_{0}^{\frac{π}{6}} - \int_{0}^{\frac{π}{6}} - \frac{\cos (3 x)}{3} d x)$

Step 7

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

$\frac{x^{2} \sin (3 x)}{3}]_{0}^{\frac{π}{6}} - \frac{2}{3} (- \frac{x \cos (3 x)}{3}]_{0}^{\frac{π}{6}} - - \int_{0}^{\frac{π}{6}} \frac{\cos (3 x)}{3} d x)$

Step 8

Simplify.

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Step 8.1

Multiply $- 1$ by $- 1$ .

$\frac{x^{2} \sin (3 x)}{3}]_{0}^{\frac{π}{6}} - \frac{2}{3} (- \frac{x \cos (3 x)}{3}]_{0}^{\frac{π}{6}} + 1 \int_{0}^{\frac{π}{6}} \frac{\cos (3 x)}{3} d x)$

Step 8.2

Multiply $\int_{0}^{\frac{π}{6}} \frac{\cos (3 x)}{3} d x$ by $1$ .

$\frac{x^{2} \sin (3 x)}{3}]_{0}^{\frac{π}{6}} - \frac{2}{3} (- \frac{x \cos (3 x)}{3}]_{0}^{\frac{π}{6}} + \int_{0}^{\frac{π}{6}} \frac{\cos (3 x)}{3} d x)$

Step 9

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

$\frac{x^{2} \sin (3 x)}{3}]_{0}^{\frac{π}{6}} - \frac{2}{3} (- \frac{x \cos (3 x)}{3}]_{0}^{\frac{π}{6}} + \frac{1}{3} \int_{0}^{\frac{π}{6}} \cos (3 x) d x)$

Step 10

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

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Step 10.1

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

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Step 10.1.1

Differentiate $3 x$ .

$\frac{d}{d x} [3 x]$

Step 10.1.2

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

$3 \frac{d}{d x} [x]$

Step 10.1.3

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

$3 \cdot 1$

Step 10.1.4

Multiply $3$ by $1$ .

$3$

Step 10.2

Substitute the lower limit in for $x$ in $u = 3 x$ .

$u_{lower} = 3 \cdot 0$

Step 10.3

Multiply $3$ by $0$ .

$u_{lower} = 0$

Step 10.4

Substitute the upper limit in for $x$ in $u = 3 x$ .

$u_{upper} = 3 \frac{π}{6}$

Step 10.5

Cancel the common factor of $3$ .

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Step 10.5.1

Factor $3$ out of $6$ .

$u_{upper} = 3 \frac{π}{3 (2)}$

Step 10.5.2

Cancel the common factor.

$u_{upper} = 3 \frac{π}{3 \cdot 2}$

Step 10.5.3

Rewrite the expression.

$u_{upper} = \frac{π}{2}$

Step 10.6

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

$u_{lower} = 0$

$u_{upper} = \frac{π}{2}$

Step 10.7

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

$\frac{x^{2} \sin (3 x)}{3}]_{0}^{\frac{π}{6}} - \frac{2}{3} (- \frac{x \cos (3 x)}{3}]_{0}^{\frac{π}{6}} + \frac{1}{3} \int_{0}^{\frac{π}{2}} \cos (u) \frac{1}{3} d u)$

Step 11

Combine $\cos (u)$ and $\frac{1}{3}$ .

$\frac{x^{2} \sin (3 x)}{3}]_{0}^{\frac{π}{6}} - \frac{2}{3} (- \frac{x \cos (3 x)}{3}]_{0}^{\frac{π}{6}} + \frac{1}{3} \int_{0}^{\frac{π}{2}} \frac{\cos (u)}{3} d u)$

Step 12

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$\frac{x^{2} \sin (3 x)}{3}]_{0}^{\frac{π}{6}} - \frac{2}{3} (- \frac{x \cos (3 x)}{3}]_{0}^{\frac{π}{6}} + \frac{1}{3} (\frac{1}{3} \int_{0}^{\frac{π}{2}} \cos (u) d u))$

Step 13

Simplify.

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Step 13.1

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

$\frac{x^{2} \sin (3 x)}{3}]_{0}^{\frac{π}{6}} - \frac{2}{3} (- \frac{x \cos (3 x)}{3}]_{0}^{\frac{π}{6}} + \frac{1}{3 \cdot 3} \int_{0}^{\frac{π}{2}} \cos (u) d u)$

Step 13.2

Multiply $3$ by $3$ .

$\frac{x^{2} \sin (3 x)}{3}]_{0}^{\frac{π}{6}} - \frac{2}{3} (- \frac{x \cos (3 x)}{3}]_{0}^{\frac{π}{6}} + \frac{1}{9} \int_{0}^{\frac{π}{2}} \cos (u) d u)$

Step 14

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

$\frac{x^{2} \sin (3 x)}{3}]_{0}^{\frac{π}{6}} - \frac{2}{3} (- \frac{x \cos (3 x)}{3}]_{0}^{\frac{π}{6}} + \frac{1}{9} \sin (u)]_{0}^{\frac{π}{2}})$

Step 15

Combine $\frac{1}{9}$ and $\sin (u)]_{0}^{\frac{π}{2}}$ .

$\frac{x^{2} \sin (3 x)}{3}]_{0}^{\frac{π}{6}} - \frac{2}{3} (- \frac{x \cos (3 x)}{3}]_{0}^{\frac{π}{6}} + \frac{\sin (u)]_{0}^{\frac{π}{2}}}{9})$

Step 16

Substitute and simplify.

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Step 16.1

Evaluate $\frac{x^{2} \sin (3 x)}{3}$ at $\frac{π}{6}$ and at $0$ .

$(\frac{{(\frac{π}{6})}^{2} \sin (3 \frac{π}{6})}{3}) - \frac{0^{2} \sin (3 \cdot 0)}{3} - \frac{2}{3} (- \frac{x \cos (3 x)}{3}]_{0}^{\frac{π}{6}} + \frac{\sin (u)]_{0}^{\frac{π}{2}}}{9})$

Step 16.2

Evaluate $- \frac{x \cos (3 x)}{3}$ at $\frac{π}{6}$ and at $0$ .

$(\frac{{(\frac{π}{6})}^{2} \sin (3 \frac{π}{6})}{3}) - \frac{0^{2} \sin (3 \cdot 0)}{3} - \frac{2}{3} ((- \frac{\frac{π}{6} \cos (3 \frac{π}{6})}{3}) + \frac{0 \cos (3 \cdot 0)}{3} + \frac{\sin (u)]_{0}^{\frac{π}{2}}}{9})$

Step 16.3

Evaluate $\sin (u)$ at $\frac{π}{2}$ and at $0$ .

Step 16.4

Simplify.

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Step 16.4.1

Combine $3$ and $\frac{π}{6}$ .

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{3 π}{6})}{3} - \frac{0^{2} \sin (3 \cdot 0)}{3} - \frac{2}{3} ((- \frac{\frac{π}{6} \cos (3 \frac{π}{6})}{3}) + \frac{0 \cos (3 \cdot 0)}{3} + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.2

Cancel the common factor of $3$ and $6$ .

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Step 16.4.2.1

Factor $3$ out of $3 π$ .

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{3 (π)}{6})}{3} - \frac{0^{2} \sin (3 \cdot 0)}{3} - \frac{2}{3} ((- \frac{\frac{π}{6} \cos (3 \frac{π}{6})}{3}) + \frac{0 \cos (3 \cdot 0)}{3} + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.2.2

Cancel the common factors.

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Step 16.4.2.2.1

Factor $3$ out of $6$ .

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{3 π}{3 \cdot 2})}{3} - \frac{0^{2} \sin (3 \cdot 0)}{3} - \frac{2}{3} ((- \frac{\frac{π}{6} \cos (3 \frac{π}{6})}{3}) + \frac{0 \cos (3 \cdot 0)}{3} + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.2.2.2

Cancel the common factor.

Step 16.4.2.2.3

Rewrite the expression.

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{0^{2} \sin (3 \cdot 0)}{3} - \frac{2}{3} ((- \frac{\frac{π}{6} \cos (3 \frac{π}{6})}{3}) + \frac{0 \cos (3 \cdot 0)}{3} + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.3

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

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{0 \sin (3 \cdot 0)}{3} - \frac{2}{3} ((- \frac{\frac{π}{6} \cos (3 \frac{π}{6})}{3}) + \frac{0 \cos (3 \cdot 0)}{3} + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.4

Multiply $3$ by $0$ .

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{0 \sin (0)}{3} - \frac{2}{3} ((- \frac{\frac{π}{6} \cos (3 \frac{π}{6})}{3}) + \frac{0 \cos (3 \cdot 0)}{3} + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.5

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

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{0}{3} - \frac{2}{3} ((- \frac{\frac{π}{6} \cos (3 \frac{π}{6})}{3}) + \frac{0 \cos (3 \cdot 0)}{3} + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.6

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

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Step 16.4.6.1

Factor $3$ out of $0$ .

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{3 (0)}{3} - \frac{2}{3} ((- \frac{\frac{π}{6} \cos (3 \frac{π}{6})}{3}) + \frac{0 \cos (3 \cdot 0)}{3} + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.6.2

Cancel the common factors.

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Step 16.4.6.2.1

Factor $3$ out of $3$ .

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{3 \cdot 0}{3 \cdot 1} - \frac{2}{3} ((- \frac{\frac{π}{6} \cos (3 \frac{π}{6})}{3}) + \frac{0 \cos (3 \cdot 0)}{3} + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.6.2.2

Cancel the common factor.

Step 16.4.6.2.3

Rewrite the expression.

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{0}{1} - \frac{2}{3} ((- \frac{\frac{π}{6} \cos (3 \frac{π}{6})}{3}) + \frac{0 \cos (3 \cdot 0)}{3} + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.6.2.4

Divide $0$ by $1$ .

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - 0 - \frac{2}{3} ((- \frac{\frac{π}{6} \cos (3 \frac{π}{6})}{3}) + \frac{0 \cos (3 \cdot 0)}{3} + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.7

Multiply $- 1$ by $0$ .

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} + 0 - \frac{2}{3} ((- \frac{\frac{π}{6} \cos (3 \frac{π}{6})}{3}) + \frac{0 \cos (3 \cdot 0)}{3} + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.8

Add $\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3}$ and $0$ .

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{2}{3} ((- \frac{\frac{π}{6} \cos (3 \frac{π}{6})}{3}) + \frac{0 \cos (3 \cdot 0)}{3} + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.9

Combine $3$ and $\frac{π}{6}$ .

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{2}{3} (- \frac{\frac{π}{6} \cos (\frac{3 π}{6})}{3} + \frac{0 \cos (3 \cdot 0)}{3} + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.10

Cancel the common factor of $3$ and $6$ .

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Step 16.4.10.1

Factor $3$ out of $3 π$ .

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{2}{3} (- \frac{\frac{π}{6} \cos (\frac{3 (π)}{6})}{3} + \frac{0 \cos (3 \cdot 0)}{3} + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.10.2

Cancel the common factors.

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Step 16.4.10.2.1

Factor $3$ out of $6$ .

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{2}{3} (- \frac{\frac{π}{6} \cos (\frac{3 π}{3 \cdot 2})}{3} + \frac{0 \cos (3 \cdot 0)}{3} + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.10.2.2

Cancel the common factor.

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{2}{3} (- \frac{\frac{π}{6} \cos (\frac{3 π}{3 \cdot 2})}{3} + \frac{0 \cos (3 \cdot 0)}{3} + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.10.2.3

Rewrite the expression.

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{2}{3} (- \frac{\frac{π}{6} \cos (\frac{π}{2})}{3} + \frac{0 \cos (3 \cdot 0)}{3} + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.11

Combine $\frac{π}{6}$ and $\cos (\frac{π}{2})$ .

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{2}{3} (- \frac{\frac{π \cos (\frac{π}{2})}{6}}{3} + \frac{0 \cos (3 \cdot 0)}{3} + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.12

Rewrite $\frac{\frac{π \cos (\frac{π}{2})}{6}}{3}$ as a product.

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{2}{3} (- (\frac{π \cos (\frac{π}{2})}{6} \cdot \frac{1}{3}) + \frac{0 \cos (3 \cdot 0)}{3} + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.13

Multiply $\frac{π \cos (\frac{π}{2})}{6}$ by $\frac{1}{3}$ .

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{2}{3} (- \frac{π \cos (\frac{π}{2})}{6 \cdot 3} + \frac{0 \cos (3 \cdot 0)}{3} + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.14

Multiply $6$ by $3$ .

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{2}{3} (- \frac{π \cos (\frac{π}{2})}{18} + \frac{0 \cos (3 \cdot 0)}{3} + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.15

Multiply $3$ by $0$ .

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{2}{3} (- \frac{π \cos (\frac{π}{2})}{18} + \frac{0 \cos (0)}{3} + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.16

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

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{2}{3} (- \frac{π \cos (\frac{π}{2})}{18} + \frac{0}{3} + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.17

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

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Step 16.4.17.1

Factor $3$ out of $0$ .

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{2}{3} (- \frac{π \cos (\frac{π}{2})}{18} + \frac{3 (0)}{3} + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.17.2

Cancel the common factors.

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Step 16.4.17.2.1

Factor $3$ out of $3$ .

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{2}{3} (- \frac{π \cos (\frac{π}{2})}{18} + \frac{3 \cdot 0}{3 \cdot 1} + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.17.2.2

Cancel the common factor.

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{2}{3} (- \frac{π \cos (\frac{π}{2})}{18} + \frac{3 \cdot 0}{3 \cdot 1} + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.17.2.3

Rewrite the expression.

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{2}{3} (- \frac{π \cos (\frac{π}{2})}{18} + \frac{0}{1} + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.17.2.4

Divide $0$ by $1$ .

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{2}{3} (- \frac{π \cos (\frac{π}{2})}{18} + 0 + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.18

Add $- \frac{π \cos (\frac{π}{2})}{18}$ and $0$ .

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{2}{3} (- \frac{π \cos (\frac{π}{2})}{18} + \frac{(\sin (\frac{π}{2})) - \sin (0)}{9})$

Step 16.4.19

To write $\frac{\sin (\frac{π}{2}) - \sin (0)}{9}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{2}{3} (- \frac{π \cos (\frac{π}{2})}{18} + \frac{\sin (\frac{π}{2}) - \sin (0)}{9} \cdot \frac{2}{2})$

Step 16.4.20

Write each expression with a common denominator of $18$ , by multiplying each by an appropriate factor of $1$ .

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Step 16.4.20.1

Multiply $\frac{\sin (\frac{π}{2}) - \sin (0)}{9}$ by $\frac{2}{2}$ .

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{2}{3} (- \frac{π \cos (\frac{π}{2})}{18} + \frac{(\sin (\frac{π}{2}) - \sin (0)) \cdot 2}{9 \cdot 2})$

Step 16.4.20.2

Multiply $9$ by $2$ .

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{2}{3} (- \frac{π \cos (\frac{π}{2})}{18} + \frac{(\sin (\frac{π}{2}) - \sin (0)) \cdot 2}{18})$

Step 16.4.21

Combine the numerators over the common denominator.

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{2}{3} \cdot \frac{- π \cos (\frac{π}{2}) + (\sin (\frac{π}{2}) - \sin (0)) \cdot 2}{18}$

Step 16.4.22

Move $2$ to the left of $\sin (\frac{π}{2}) - \sin (0)$ .

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{2}{3} \cdot \frac{- π \cos (\frac{π}{2}) + 2 \cdot (\sin (\frac{π}{2}) - \sin (0))}{18}$

Step 16.4.23

Multiply $\frac{- π \cos (\frac{π}{2}) + 2 (\sin (\frac{π}{2}) - \sin (0))}{18}$ by $\frac{2}{3}$ .

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{(- π \cos (\frac{π}{2}) + 2 (\sin (\frac{π}{2}) - \sin (0))) \cdot 2}{18 \cdot 3}$

Step 16.4.24

Multiply $18$ by $3$ .

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{(- π \cos (\frac{π}{2}) + 2 (\sin (\frac{π}{2}) - \sin (0))) \cdot 2}{54}$

Step 16.4.25

Move $2$ to the left of $- π \cos (\frac{π}{2}) + 2 (\sin (\frac{π}{2}) - \sin (0))$ .

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{2 \cdot (- π \cos (\frac{π}{2}) + 2 (\sin (\frac{π}{2}) - \sin (0)))}{54}$

Step 16.4.26

Cancel the common factors.

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Step 16.4.26.1

Factor $2$ out of $54$ .

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{2 (- π \cos (\frac{π}{2}) + 2 (\sin (\frac{π}{2}) - \sin (0)))}{2 \cdot 27}$

Step 16.4.26.2

Cancel the common factor.

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{2 (- π \cos (\frac{π}{2}) + 2 (\sin (\frac{π}{2}) - \sin (0)))}{2 \cdot 27}$

Step 16.4.26.3

Rewrite the expression.

$\frac{{(\frac{π}{6})}^{2} \sin (\frac{π}{2})}{3} - \frac{- π \cos (\frac{π}{2}) + 2 (\sin (\frac{π}{2}) - \sin (0))}{27}$

Step 17

Simplify.

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Step 17.1

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$\frac{{(\frac{π}{6})}^{2} \cdot 1}{3} - \frac{- π \cos (\frac{π}{2}) + 2 (\sin (\frac{π}{2}) - \sin (0))}{27}$

Step 17.2

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$\frac{{(\frac{π}{6})}^{2} \cdot 1}{3} - \frac{- π \cdot 0 + 2 (\sin (\frac{π}{2}) - \sin (0))}{27}$

Step 17.3

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$\frac{{(\frac{π}{6})}^{2} \cdot 1}{3} - \frac{- π \cdot 0 + 2 (1 - \sin (0))}{27}$

Step 17.4

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

$\frac{{(\frac{π}{6})}^{2} \cdot 1}{3} - \frac{- π \cdot 0 + 2 (1 - 0)}{27}$

Step 17.5

Multiply ${(\frac{π}{6})}^{2}$ by $1$ .

$\frac{{(\frac{π}{6})}^{2}}{3} - \frac{- π \cdot 0 + 2 (1 - 0)}{27}$

Step 17.6

Multiply $0$ by $- 1$ .

$\frac{{(\frac{π}{6})}^{2}}{3} - \frac{0 π + 2 (1 - 0)}{27}$

Step 17.7

Multiply $0$ by $π$ .

$\frac{{(\frac{π}{6})}^{2}}{3} - \frac{0 + 2 (1 - 0)}{27}$

Step 17.8

Multiply $- 1$ by $0$ .

$\frac{{(\frac{π}{6})}^{2}}{3} - \frac{0 + 2 (1 + 0)}{27}$

Step 17.9

Add $1$ and $0$ .

$\frac{{(\frac{π}{6})}^{2}}{3} - \frac{0 + 2 \cdot 1}{27}$

Step 17.10

Multiply $2$ by $1$ .

$\frac{{(\frac{π}{6})}^{2}}{3} - \frac{0 + 2}{27}$

Step 17.11

Add $0$ and $2$ .

$\frac{{(\frac{π}{6})}^{2}}{3} - \frac{2}{27}$

Step 18

Simplify.

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Step 18.1

Simplify the numerator.

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Step 18.1.1

Apply the product rule to $\frac{π}{6}$ .

$\frac{\frac{π^{2}}{6^{2}}}{3} - \frac{2}{27}$

Step 18.1.2

Raise $6$ to the power of $2$ .

$\frac{\frac{π^{2}}{36}}{3} - \frac{2}{27}$

Step 18.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{π^{2}}{36} \cdot \frac{1}{3} - \frac{2}{27}$

Step 18.3

Combine.

$\frac{π^{2} \cdot 1}{36 \cdot 3} - \frac{2}{27}$

Step 18.4

Multiply $π^{2}$ by $1$ .

$\frac{π^{2}}{36 \cdot 3} - \frac{2}{27}$

Step 18.5

Multiply $36$ by $3$ .

$\frac{π^{2}}{108} - \frac{2}{27}$

Step 19

The result can be shown in multiple forms.

Exact Form:

$\frac{π^{2}}{108} - \frac{2}{27}$

Decimal Form:

$0.01731115 \dots$