Calculus Examples

$\int_{0}^{\frac{π}{4}} x \sin (2 x) d x$

Step 1

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

$x (- \frac{1}{2} \cos (2 x))]_{0}^{\frac{π}{4}} - \int_{0}^{\frac{π}{4}} - \frac{1}{2} \cos (2 x) d x$

Step 2

Simplify.

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Step 2.1

Combine $\cos (2 x)$ and $\frac{1}{2}$ .

$x (- \frac{\cos (2 x)}{2})]_{0}^{\frac{π}{4}} - \int_{0}^{\frac{π}{4}} - \frac{1}{2} \cos (2 x) d x$

Step 2.2

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

$- \frac{x \cos (2 x)}{2}]_{0}^{\frac{π}{4}} - \int_{0}^{\frac{π}{4}} - \frac{1}{2} \cos (2 x) d x$

Step 3

Since $- \frac{1}{2}$ is constant with respect to $x$ , move $- \frac{1}{2}$ out of the integral.

$- \frac{x \cos (2 x)}{2}]_{0}^{\frac{π}{4}} - (- \frac{1}{2} \int_{0}^{\frac{π}{4}} \cos (2 x) d x)$

Step 4

Simplify.

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Step 4.1

Multiply $- 1$ by $- 1$ .

$- \frac{x \cos (2 x)}{2}]_{0}^{\frac{π}{4}} + 1 (\frac{1}{2} \int_{0}^{\frac{π}{4}} \cos (2 x) d x)$

Step 4.2

Multiply $\frac{1}{2}$ by $1$ .

$- \frac{x \cos (2 x)}{2}]_{0}^{\frac{π}{4}} + \frac{1}{2} \int_{0}^{\frac{π}{4}} \cos (2 x) d x$

Step 5

Let $u = 2 x$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = 2 x$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 5.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 5.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 5.1.4

Multiply $2$ by $1$ .

$2$

Step 5.2

Substitute the lower limit in for $x$ in $u = 2 x$ .

$u_{lower} = 2 \cdot 0$

Step 5.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 5.4

Substitute the upper limit in for $x$ in $u = 2 x$ .

$u_{upper} = 2 \frac{π}{4}$

Step 5.5

Cancel the common factor of $2$ .

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Step 5.5.1

Factor $2$ out of $4$ .

$u_{upper} = 2 \frac{π}{2 (2)}$

Step 5.5.2

Cancel the common factor.

$u_{upper} = 2 \frac{π}{2 \cdot 2}$

Step 5.5.3

Rewrite the expression.

$u_{upper} = \frac{π}{2}$

Step 5.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = \frac{π}{2}$

Step 5.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$- \frac{x \cos (2 x)}{2}]_{0}^{\frac{π}{4}} + \frac{1}{2} \int_{0}^{\frac{π}{2}} \cos (u) \frac{1}{2} d u$

Step 6

Combine $\cos (u)$ and $\frac{1}{2}$ .

$- \frac{x \cos (2 x)}{2}]_{0}^{\frac{π}{4}} + \frac{1}{2} \int_{0}^{\frac{π}{2}} \frac{\cos (u)}{2} d u$

Step 7

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$- \frac{x \cos (2 x)}{2}]_{0}^{\frac{π}{4}} + \frac{1}{2} (\frac{1}{2} \int_{0}^{\frac{π}{2}} \cos (u) d u)$

Step 8

Simplify.

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Step 8.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$- \frac{x \cos (2 x)}{2}]_{0}^{\frac{π}{4}} + \frac{1}{2 \cdot 2} \int_{0}^{\frac{π}{2}} \cos (u) d u$

Step 8.2

Multiply $2$ by $2$ .

$- \frac{x \cos (2 x)}{2}]_{0}^{\frac{π}{4}} + \frac{1}{4} \int_{0}^{\frac{π}{2}} \cos (u) d u$

Step 9

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

$- \frac{x \cos (2 x)}{2}]_{0}^{\frac{π}{4}} + \frac{1}{4} \sin (u)]_{0}^{\frac{π}{2}}$

Step 10

Substitute and simplify.

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Step 10.1

Evaluate $- \frac{x \cos (2 x)}{2}$ at $\frac{π}{4}$ and at $0$ .

$(- \frac{\frac{π}{4} \cos (2 \frac{π}{4})}{2}) + \frac{0 \cos (2 \cdot 0)}{2} + \frac{1}{4} \sin (u)]_{0}^{\frac{π}{2}}$

Step 10.2

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

$(- \frac{\frac{π}{4} \cos (2 \frac{π}{4})}{2}) + \frac{0 \cos (2 \cdot 0)}{2} + \frac{1}{4} ((\sin (\frac{π}{2})) - \sin (0))$

Step 10.3

Simplify.

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Step 10.3.1

Combine $2$ and $\frac{π}{4}$ .

$- \frac{\frac{π}{4} \cos (\frac{2 π}{4})}{2} + \frac{0 \cos (2 \cdot 0)}{2} + \frac{1}{4} ((\sin (\frac{π}{2})) - \sin (0))$

Step 10.3.2

Cancel the common factor of $2$ and $4$ .

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Step 10.3.2.1

Factor $2$ out of $2 π$ .

$- \frac{\frac{π}{4} \cos (\frac{2 (π)}{4})}{2} + \frac{0 \cos (2 \cdot 0)}{2} + \frac{1}{4} ((\sin (\frac{π}{2})) - \sin (0))$

Step 10.3.2.2

Cancel the common factors.

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Step 10.3.2.2.1

Factor $2$ out of $4$ .

$- \frac{\frac{π}{4} \cos (\frac{2 π}{2 \cdot 2})}{2} + \frac{0 \cos (2 \cdot 0)}{2} + \frac{1}{4} ((\sin (\frac{π}{2})) - \sin (0))$

Step 10.3.2.2.2

Cancel the common factor.

$- \frac{\frac{π}{4} \cos (\frac{2 π}{2 \cdot 2})}{2} + \frac{0 \cos (2 \cdot 0)}{2} + \frac{1}{4} ((\sin (\frac{π}{2})) - \sin (0))$

Step 10.3.2.2.3

Rewrite the expression.

$- \frac{\frac{π}{4} \cos (\frac{π}{2})}{2} + \frac{0 \cos (2 \cdot 0)}{2} + \frac{1}{4} ((\sin (\frac{π}{2})) - \sin (0))$

Step 10.3.3

Combine $\frac{π}{4}$ and $\cos (\frac{π}{2})$ .

$- \frac{\frac{π \cos (\frac{π}{2})}{4}}{2} + \frac{0 \cos (2 \cdot 0)}{2} + \frac{1}{4} ((\sin (\frac{π}{2})) - \sin (0))$

Step 10.3.4

Rewrite $\frac{\frac{π \cos (\frac{π}{2})}{4}}{2}$ as a product.

$- (\frac{π \cos (\frac{π}{2})}{4} \cdot \frac{1}{2}) + \frac{0 \cos (2 \cdot 0)}{2} + \frac{1}{4} ((\sin (\frac{π}{2})) - \sin (0))$

Step 10.3.5

Multiply $\frac{π \cos (\frac{π}{2})}{4}$ by $\frac{1}{2}$ .

$- \frac{π \cos (\frac{π}{2})}{4 \cdot 2} + \frac{0 \cos (2 \cdot 0)}{2} + \frac{1}{4} ((\sin (\frac{π}{2})) - \sin (0))$

Step 10.3.6

Multiply $4$ by $2$ .

$- \frac{π \cos (\frac{π}{2})}{8} + \frac{0 \cos (2 \cdot 0)}{2} + \frac{1}{4} ((\sin (\frac{π}{2})) - \sin (0))$

Step 10.3.7

Multiply $2$ by $0$ .

$- \frac{π \cos (\frac{π}{2})}{8} + \frac{0 \cos (0)}{2} + \frac{1}{4} ((\sin (\frac{π}{2})) - \sin (0))$

Step 10.3.8

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

$- \frac{π \cos (\frac{π}{2})}{8} + \frac{0}{2} + \frac{1}{4} ((\sin (\frac{π}{2})) - \sin (0))$

Step 10.3.9

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

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Step 10.3.9.1

Factor $2$ out of $0$ .

$- \frac{π \cos (\frac{π}{2})}{8} + \frac{2 (0)}{2} + \frac{1}{4} ((\sin (\frac{π}{2})) - \sin (0))$

Step 10.3.9.2

Cancel the common factors.

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Step 10.3.9.2.1

Factor $2$ out of $2$ .

$- \frac{π \cos (\frac{π}{2})}{8} + \frac{2 \cdot 0}{2 \cdot 1} + \frac{1}{4} ((\sin (\frac{π}{2})) - \sin (0))$

Step 10.3.9.2.2

Cancel the common factor.

$- \frac{π \cos (\frac{π}{2})}{8} + \frac{2 \cdot 0}{2 \cdot 1} + \frac{1}{4} ((\sin (\frac{π}{2})) - \sin (0))$

Step 10.3.9.2.3

Rewrite the expression.

$- \frac{π \cos (\frac{π}{2})}{8} + \frac{0}{1} + \frac{1}{4} ((\sin (\frac{π}{2})) - \sin (0))$

Step 10.3.9.2.4

Divide $0$ by $1$ .

$- \frac{π \cos (\frac{π}{2})}{8} + 0 + \frac{1}{4} ((\sin (\frac{π}{2})) - \sin (0))$

Step 10.3.10

Add $- \frac{π \cos (\frac{π}{2})}{8}$ and $0$ .

$- \frac{π \cos (\frac{π}{2})}{8} + \frac{1}{4} ((\sin (\frac{π}{2})) - \sin (0))$

Step 10.3.11

To write $\frac{1}{4} (\sin (\frac{π}{2}) - \sin (0))$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$- \frac{π \cos (\frac{π}{2})}{8} + \frac{1}{4} (\sin (\frac{π}{2}) - \sin (0)) \cdot \frac{8}{8}$

Step 10.3.12

Combine $\frac{1}{4} (\sin (\frac{π}{2}) - \sin (0))$ and $\frac{8}{8}$ .

$- \frac{π \cos (\frac{π}{2})}{8} + \frac{\frac{1}{4} (\sin (\frac{π}{2}) - \sin (0)) \cdot 8}{8}$

Step 10.3.13

Combine the numerators over the common denominator.

$\frac{- π \cos (\frac{π}{2}) + \frac{1}{4} (\sin (\frac{π}{2}) - \sin (0)) \cdot 8}{8}$

Step 10.3.14

Combine $8$ and $\frac{1}{4}$ .

$\frac{- π \cos (\frac{π}{2}) + \frac{8}{4} (\sin (\frac{π}{2}) - \sin (0))}{8}$

Step 10.3.15

Cancel the common factor of $8$ and $4$ .

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Step 10.3.15.1

Factor $4$ out of $8$ .

$\frac{- π \cos (\frac{π}{2}) + \frac{4 \cdot 2}{4} (\sin (\frac{π}{2}) - \sin (0))}{8}$

Step 10.3.15.2

Cancel the common factors.

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Step 10.3.15.2.1

Factor $4$ out of $4$ .

$\frac{- π \cos (\frac{π}{2}) + \frac{4 \cdot 2}{4 (1)} (\sin (\frac{π}{2}) - \sin (0))}{8}$

Step 10.3.15.2.2

Cancel the common factor.

$\frac{- π \cos (\frac{π}{2}) + \frac{4 \cdot 2}{4 \cdot 1} (\sin (\frac{π}{2}) - \sin (0))}{8}$

Step 10.3.15.2.3

Rewrite the expression.

$\frac{- π \cos (\frac{π}{2}) + \frac{2}{1} (\sin (\frac{π}{2}) - \sin (0))}{8}$

Step 10.3.15.2.4

Divide $2$ by $1$ .

$\frac{- π \cos (\frac{π}{2}) + 2 (\sin (\frac{π}{2}) - \sin (0))}{8}$

Step 11

Simplify.

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Step 11.1

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

$\frac{- π \cdot 0 + 2 (\sin (\frac{π}{2}) - \sin (0))}{8}$

Step 11.2

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

$\frac{- π \cdot 0 + 2 (1 - \sin (0))}{8}$

Step 11.3

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

$\frac{- π \cdot 0 + 2 (1 - 0)}{8}$

Step 11.4

Multiply $0$ by $- 1$ .

$\frac{0 π + 2 (1 - 0)}{8}$

Step 11.5

Multiply $0$ by $π$ .

$\frac{0 + 2 (1 - 0)}{8}$

Step 11.6

Multiply $- 1$ by $0$ .

$\frac{0 + 2 (1 + 0)}{8}$

Step 11.7

Add $1$ and $0$ .

$\frac{0 + 2 \cdot 1}{8}$

Step 11.8

Multiply $2$ by $1$ .

$\frac{0 + 2}{8}$

Step 11.9

Add $0$ and $2$ .

$\frac{2}{8}$

Step 11.10

Cancel the common factor of $2$ and $8$ .

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Step 11.10.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{8}$

Step 11.10.2

Cancel the common factors.

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Step 11.10.2.1

Factor $2$ out of $8$ .

$\frac{2 \cdot 1}{2 \cdot 4}$

Step 11.10.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 4}$

Step 11.10.2.3

Rewrite the expression.

$\frac{1}{4}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{4}$

Decimal Form:

$0.25$