Calculus Examples

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

Step 1

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

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

Step 2

Multiply $2$ by $- 1$ .

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

Step 3

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

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

Step 4

Multiply $- 2$ by $- 1$ .

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify the answer.

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Step 7.1

Substitute and simplify.

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Step 7.1.1

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

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

Step 7.1.2

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

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

Step 7.1.3

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

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

Step 7.1.4

Simplify.

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Step 7.1.4.1

Raising $0$ to any positive power yields $0$ .

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

Step 7.1.4.2

Multiply $- 1$ by $0$ .

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

Step 7.1.4.3

Multiply $- 1$ by $0$ .

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

Step 7.1.4.4

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

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

Step 7.1.4.5

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

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

Step 7.1.4.6

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

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

Step 7.1.4.7

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

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

Step 7.1.4.8

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

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

Step 7.1.4.9

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

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

Step 7.1.4.10

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

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

Step 7.1.4.11

Combine the numerators over the common denominator.

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

Step 7.1.4.12

Multiply $2$ by $- 1$ .

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

Step 7.1.4.13

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

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

Step 7.1.4.14

Cancel the common factor of $2$ .

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Step 7.1.4.14.1

Cancel the common factor.

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

Step 7.1.4.14.2

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

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

Step 7.2

Simplify.

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Step 7.2.1

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

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

Step 7.2.2

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

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

Step 7.2.3

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

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

Step 7.2.4

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

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

Step 7.2.5

Multiply $- 1$ by $0$ .

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

Step 7.2.6

Multiply ${(\frac{π}{2})}^{2}$ by $0$ .

$0 + π \cdot 1 - 2 (- 0 + 1)$

Step 7.2.7

Multiply $π$ by $1$ .

$0 + π - 2 (- 0 + 1)$

Step 7.2.8

Add $0$ and $π$ .

$π - 2 (- 0 + 1)$

Step 7.2.9

Multiply $- 1$ by $0$ .

$π - 2 (0 + 1)$

Step 7.2.10

Add $0$ and $1$ .

$π - 2 \cdot 1$

Step 7.2.11

Multiply $- 2$ by $1$ .

$π - 2$

Step 8

The result can be shown in multiple forms.

Exact Form:

$π - 2$

Decimal Form:

$1.14159265 \dots$