Calculus Examples

$\int_{0}^{2 π} x \sin (x) d x$

Step 1

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

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.2

Multiply $\int_{0}^{2 π} \cos (x) d x$ by $1$ .

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

Step 4

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

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

Step 5

Simplify the answer.

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Step 5.1

Combine $x (- \cos (x))]_{0}^{2 π}$ and $\sin (x)]_{0}^{2 π}$ .

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

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $x (- \cos (x)) + \sin (x)$ at $2 π$ and at $0$ .

$(2 π (- \cos (2 π)) + \sin (2 π)) - (0 (- \cos (0)) + \sin (0))$

Step 5.2.2

Simplify.

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Step 5.2.2.1

Multiply $- 1$ by $2$ .

$- 2 π \cos (2 π) + \sin (2 π) - (0 (- \cos (0)) + \sin (0))$

Step 5.2.2.2

Multiply $- 1$ by $0$ .

$- 2 π \cos (2 π) + \sin (2 π) - (0 \cos (0) + \sin (0))$

Step 5.2.2.3

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

$- 2 π \cos (2 π) + \sin (2 π) - (0 + \sin (0))$

Step 5.2.2.4

Add $0$ and $\sin (0)$ .

$- 2 π \cos (2 π) + \sin (2 π) - \sin (0)$

Step 5.3

Simplify.

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Step 5.3.1

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

$- 2 π \cos (2 π) + \sin (2 π) - 0$

Step 5.3.2

Multiply $- 1$ by $0$ .

$- 2 π \cos (2 π) + \sin (2 π) + 0$

Step 5.3.3

Add $- 2 π \cos (2 π) + \sin (2 π)$ and $0$ .

$- 2 π \cos (2 π) + \sin (2 π)$

Step 5.4

Simplify.

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Step 5.4.1

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

$- 2 π \cos (0) + \sin (2 π)$

Step 5.4.2

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

$- 2 π \cdot 1 + \sin (2 π)$

Step 5.4.3

Multiply $- 2$ by $1$ .

$- 2 π + \sin (2 π)$

Step 6

Simplify.

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Step 6.1

Simplify each term.

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Step 6.1.1

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

$- 2 π + \sin (0)$

Step 6.1.2

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

$- 2 π + 0$

Step 6.2

Add $- 2 π$ and $0$ .

$- 2 π$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- 2 π$

Decimal Form:

$- 6.28318530 \dots$