Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.2

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

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

Step 4

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

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

Step 5

Simplify the answer.

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Step 5.1

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

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

Step 5.2

Substitute and simplify.

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Step 5.2.1

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

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

Step 5.2.2

Simplify.

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Step 5.2.2.1

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

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

Step 5.2.2.2

Multiply $- 1$ by $0$ .

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

Step 5.2.2.3

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

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

Step 5.2.2.4

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

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

Step 5.3

Simplify.

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Step 5.3.1

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

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

Step 5.3.2

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

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

Step 5.3.3

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

$- \frac{π \cdot 0}{2} + 1 - 0$

Step 5.3.4

Multiply $π$ by $0$ .

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

Step 5.3.5

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

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Step 5.3.5.1

Factor $2$ out of $0$ .

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

Step 5.3.5.2

Cancel the common factors.

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Step 5.3.5.2.1

Factor $2$ out of $2$ .

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

Step 5.3.5.2.2

Cancel the common factor.

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

Step 5.3.5.2.3

Rewrite the expression.

$- \frac{0}{1} + 1 - 0$

Step 5.3.5.2.4

Divide $0$ by $1$ .

$- 0 + 1 - 0$

Step 5.3.6

Multiply $- 1$ by $0$ .

$0 + 1 - 0$

Step 5.3.7

Add $0$ and $1$ .

$1 - 0$

Step 5.3.8

Multiply $- 1$ by $0$ .

$1 + 0$

Step 5.3.9

Add $1$ and $0$ .

$1$