Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Substitute and simplify.

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

Simplify.

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Step 3.1.3.1

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

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

Step 3.1.3.2

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

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

Step 3.1.3.3

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

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

Step 3.1.3.4

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

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

Step 3.1.3.5

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

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

Step 3.1.3.6

Combine the numerators over the common denominator.

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

Step 3.1.3.7

Multiply $2$ by $- 1$ .

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

Step 3.2

Simplify.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

$\frac{π \cdot 1 - 2 (- 0 + 1)}{2}$

Step 3.2.4

Multiply $π$ by $1$ .

$\frac{π - 2 (- 0 + 1)}{2}$

Step 3.2.5

Multiply $- 1$ by $0$ .

$\frac{π - 2 (0 + 1)}{2}$

Step 3.2.6

Add $0$ and $1$ .

$\frac{π - 2 \cdot 1}{2}$

Step 3.2.7

Multiply $- 2$ by $1$ .

$\frac{π - 2}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{π - 2}{2}$

Decimal Form:

$0.57079632 \dots$