Calculus Examples

$\int_{0}^{π} x \sin (2 x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \sin (2 x)$ .

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

Step 2

Simplify.

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Step 2.1

Combine $\cos (2 x)$ and $\frac{1}{2}$ .

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

Step 2.2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.2

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

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

Step 5

Let $u = 2 x$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = 2 x$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 5.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 5.1.3

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

$2 \cdot 1$

Step 5.1.4

Multiply $2$ by $1$ .

$2$

Step 5.2

Substitute the lower limit in for $x$ in $u = 2 x$ .

$u_{lower} = 2 \cdot 0$

Step 5.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 5.4

Substitute the upper limit in for $x$ in $u = 2 x$ .

$u_{upper} = 2 π$

Step 5.5

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

$u_{lower} = 0$

$u_{upper} = 2 π$

Step 5.6

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

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

Step 6

Combine $\cos (u)$ and $\frac{1}{2}$ .

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

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

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

Step 8.2

Multiply $2$ by $2$ .

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

Step 9

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

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

Step 10

Substitute and simplify.

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

Simplify.

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Step 10.3.1

Multiply $2$ by $0$ .

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

Step 10.3.2

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

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

Step 10.3.3

Cancel the common factor of $0$ and $2$ .

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Step 10.3.3.1

Factor $2$ out of $0$ .

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

Step 10.3.3.2

Cancel the common factors.

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Step 10.3.3.2.1

Factor $2$ out of $2$ .

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

Step 10.3.3.2.2

Cancel the common factor.

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

Step 10.3.3.2.3

Rewrite the expression.

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

Step 10.3.3.2.4

Divide $0$ by $1$ .

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

Step 10.3.4

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

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

Step 10.3.5

To write $\frac{1}{4} (\sin (2 π) - \sin (0))$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 10.3.6

Combine $\frac{1}{4} (\sin (2 π) - \sin (0))$ and $\frac{2}{2}$ .

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

Step 10.3.7

Combine the numerators over the common denominator.

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

Step 10.3.8

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

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

Step 10.3.9

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

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Step 10.3.9.1

Factor $2$ out of $2$ .

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

Step 10.3.9.2

Cancel the common factors.

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Step 10.3.9.2.1

Factor $2$ out of $4$ .

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

Step 10.3.9.2.2

Cancel the common factor.

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

Step 10.3.9.2.3

Rewrite the expression.

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

Step 11

Simplify.

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Step 11.1

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

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

Step 11.2

Multiply $- 1$ by $0$ .

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

Step 11.3

Add $\sin (2 π)$ and $0$ .

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

Step 11.4

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

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

Step 11.5

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

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

Step 11.6

Combine.

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

Step 11.7

Apply the distributive property.

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

Step 11.8

Cancel the common factor of $2$ .

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Step 11.8.1

Cancel the common factor.

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

Step 11.8.2

Rewrite the expression.

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

Step 11.9

Multiply $- 1$ by $2$ .

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

Step 11.10

Multiply $2$ by $2$ .

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

Step 11.11

Factor $- 1$ out of $- 2 π \cos (2 π)$ .

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

Step 11.12

Factor $- 1$ out of $\sin (2 π)$ .

$\frac{- (2 π \cos (2 π)) - 1 (- \sin (2 π))}{4}$

Step 11.13

Factor $- 1$ out of $- (2 π \cos (2 π)) - 1 (- \sin (2 π))$ .

$\frac{- (2 π \cos (2 π) - \sin (2 π))}{4}$

Step 11.14

Rewrite $- (2 π \cos (2 π) - \sin (2 π))$ as $- 1 (2 π \cos (2 π) - \sin (2 π))$ .

$\frac{- 1 (2 π \cos (2 π) - \sin (2 π))}{4}$

Step 11.15

Move the negative in front of the fraction.

$- \frac{2 π \cos (2 π) - \sin (2 π)}{4}$

Step 12

Simplify.

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Step 12.1

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

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

Step 12.2

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

$- \frac{2 π \cdot 1 - \sin (2 π)}{4}$

Step 12.3

Multiply $2$ by $1$ .

$- \frac{2 π - \sin (2 π)}{4}$

Step 12.4

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

$- \frac{2 π - \sin (0)}{4}$

Step 12.5

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

$- \frac{2 π - 0}{4}$

Step 12.6

Multiply $- 1$ by $0$ .

$- \frac{2 π + 0}{4}$

Step 12.7

Cancel the common factor of $2 π + 0$ and $4$ .

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Step 12.7.1

Factor $2$ out of $2 π$ .

$- \frac{2 (π) + 0}{4}$

Step 12.7.2

Factor $2$ out of $0$ .

$- \frac{2 (π) + 2 (0)}{4}$

Step 12.7.3

Factor $2$ out of $2 (π) + 2 (0)$ .

$- \frac{2 (π + 0)}{4}$

Step 12.7.4

Cancel the common factors.

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Step 12.7.4.1

Factor $2$ out of $4$ .

$- \frac{2 (π + 0)}{2 (2)}$

Step 12.7.4.2

Cancel the common factor.

$- \frac{2 (π + 0)}{2 \cdot 2}$

Step 12.7.4.3

Rewrite the expression.

$- \frac{π + 0}{2}$

Step 12.8

Add $π$ and $0$ .

$- \frac{π}{2}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$- \frac{π}{2}$

Decimal Form:

$- 1.57079632 \dots$