Calculus Examples

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

Step 1

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

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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Step 4.1.1

Differentiate $π x$ .

$\frac{d}{d x} [π x]$

Step 4.1.2

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

$π \frac{d}{d x} [x]$

Step 4.1.3

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

$π \cdot 1$

Step 4.1.4

Multiply $π$ by $1$ .

$π$

Step 4.2

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

$u_{lower} = π \cdot 0$

Step 4.3

Multiply $π$ by $0$ .

$u_{lower} = 0$

Step 4.4

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

$u_{upper} = π \frac{1}{2}$

Step 4.5

Combine $π$ and $\frac{1}{2}$ .

$u_{upper} = \frac{π}{2}$

Step 4.6

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 4.7

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

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

Step 5

Combine $\sin (u)$ and $\frac{1}{π}$ .

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

Raise $π$ to the power of $1$ .

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

Step 7.3

Raise $π$ to the power of $1$ .

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

Step 7.4

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

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

Step 7.5

Add $1$ and $1$ .

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

Step 8

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

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

Step 9

Substitute and simplify.

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Step 9.1

Evaluate $\frac{x \sin (π x)}{π}$ at $\frac{1}{2}$ and at $0$ .

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

Step 9.2

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

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

Step 9.3

Simplify.

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Step 9.3.1

Combine $π$ and $\frac{1}{2}$ .

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

Step 9.3.2

Combine $\frac{1}{2}$ and $\sin (\frac{π}{2})$ .

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

Step 9.3.3

Rewrite $\frac{\frac{\sin (\frac{π}{2})}{2}}{π}$ as a product.

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

Step 9.3.4

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

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

Step 9.3.5

Multiply $π$ by $0$ .

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

Step 9.3.6

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

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

Step 9.3.7

To write $- \frac{1}{π^{2}} (- \cos (\frac{π}{2}) + \cos (0))$ as a fraction with a common denominator, multiply by $\frac{2 π}{2 π}$ .

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

Step 9.3.8

Combine $- \frac{1}{π^{2}} (- \cos (\frac{π}{2}) + \cos (0))$ and $\frac{2 π}{2 π}$ .

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

Step 9.3.9

Combine the numerators over the common denominator.

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

Step 9.3.10

Multiply $2$ by $- 1$ .

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

Step 9.3.11

Combine $- 2$ and $\frac{1}{π^{2}}$ .

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

Step 9.3.12

Combine $π$ and $\frac{- 2}{π^{2}}$ .

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

Step 9.3.13

Move $- 2$ to the left of $π$ .

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

Step 9.3.14

Cancel the common factor of $π$ and $π^{2}$ .

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Step 9.3.14.1

Factor $π$ out of $- 2 π$ .

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

Step 9.3.14.2

Cancel the common factors.

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Step 9.3.14.2.1

Factor $π$ out of $π^{2}$ .

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

Step 9.3.14.2.2

Cancel the common factor.

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

Step 9.3.14.2.3

Rewrite the expression.

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

Step 9.3.15

Move the negative in front of the fraction.

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

Step 10

Simplify.

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Step 10.1

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

$\frac{1 - \frac{2}{π} (- \cos (\frac{π}{2}) + \cos (0))}{2 π} - \frac{0}{π}$

Step 10.2

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

$\frac{1 - \frac{2}{π} (- 0 + \cos (0))}{2 π} - \frac{0}{π}$

Step 10.3

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

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

Step 10.4

Multiply $- 1$ by $0$ .

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

Step 10.5

Add $0$ and $1$ .

$\frac{1 - \frac{2}{π} \cdot 1}{2 π} - \frac{0}{π}$

Step 10.6

Multiply $- 1$ by $1$ .

$\frac{1 - \frac{2}{π}}{2 π} - \frac{0}{π}$

Step 10.7

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

$\frac{1 - \frac{2}{π}}{2 π} - \frac{0}{π} \cdot \frac{2}{2}$

Step 10.8

Write each expression with a common denominator of $2 π$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.8.1

Multiply $\frac{0}{π}$ by $\frac{2}{2}$ .

$\frac{1 - \frac{2}{π}}{2 π} - \frac{0 \cdot 2}{π \cdot 2}$

Step 10.8.2

Reorder the factors of $π \cdot 2$ .

$\frac{1 - \frac{2}{π}}{2 π} - \frac{0 \cdot 2}{2 π}$

Step 10.9

Combine the numerators over the common denominator.

$\frac{1 - \frac{2}{π} + 0 \cdot 2}{2 π}$

Step 10.10

Multiply $0$ by $2$ .

$\frac{1 - \frac{2}{π} + 0}{2 π}$

Step 10.11

Add $1 - \frac{2}{π}$ and $0$ .

$\frac{1 - \frac{2}{π}}{2 π}$

Step 11

Simplify.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

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

$\frac{\frac{π}{π} - \frac{2}{π}}{2 π}$

Step 11.1.2

Combine the numerators over the common denominator.

$\frac{\frac{π - 2}{π}}{2 π}$

Step 11.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{π - 2}{π} \cdot \frac{1}{2 π}$

Step 11.3

Multiply $\frac{π - 2}{π} \cdot \frac{1}{2 π}$ .

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Step 11.3.1

Multiply $\frac{π - 2}{π}$ by $\frac{1}{2 π}$ .

$\frac{π - 2}{π (2 π)}$

Step 11.3.2

Raise $π$ to the power of $1$ .

$\frac{π - 2}{2 (π^{1} π)}$

Step 11.3.3

Raise $π$ to the power of $1$ .

$\frac{π - 2}{2 (π^{1} π^{1})}$

Step 11.3.4

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

$\frac{π - 2}{2 π^{1 + 1}}$

Step 11.3.5

Add $1$ and $1$ .

$\frac{π - 2}{2 π^{2}}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{π - 2}{2 π^{2}}$

Decimal Form:

$0.05783375 \dots$