Calculus Examples

$\int_{\frac{π}{2}}^{2 π} x \sin (x) d x$

Step 1

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

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.2

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

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

Step 4

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

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

Step 5

Simplify the answer.

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Step 5.1

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

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

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 $\frac{π}{2}$ .

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

Step 5.2.2

Simplify.

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Step 5.2.2.1

Multiply $- 1$ by $2$ .

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

Step 5.2.2.2

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

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

Step 5.3

Simplify.

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Step 5.3.1

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

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

Step 5.3.2

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

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

Step 5.3.3

Multiply $π$ by $0$ .

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

Step 5.3.4

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

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Step 5.3.4.1

Factor $2$ out of $0$ .

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

Step 5.3.4.2

Cancel the common factors.

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Step 5.3.4.2.1

Factor $2$ out of $2$ .

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

Step 5.3.4.2.2

Cancel the common factor.

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

Step 5.3.4.2.3

Rewrite the expression.

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

Step 5.3.4.2.4

Divide $0$ by $1$ .

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

Step 5.3.5

Multiply $- 1$ by $0$ .

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

Step 5.3.6

Add $0$ and $1$ .

$- 2 π \cos (2 π) + \sin (2 π) - 1 \cdot 1$

Step 5.3.7

Multiply $- 1$ by $1$ .

$- 2 π \cos (2 π) + \sin (2 π) - 1$

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 $2 π$ .

$- 2 π \cos (0) + \sin (2 π) - 1$

Step 5.4.2

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

$- 2 π \cdot 1 + \sin (2 π) - 1$

Step 5.4.3

Multiply $- 2$ by $1$ .

$- 2 π + \sin (2 π) - 1$

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 $2 π$ .

$- 2 π + \sin (0) - 1$

Step 6.1.2

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

$- 2 π + 0 - 1$

Step 6.2

Add $- 2 π$ and $0$ .

$- 2 π - 1$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- 2 π - 1$

Decimal Form:

$- 7.28318530 \dots$