Calculus Examples

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

Step 1

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

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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

Step 4

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 4.1

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

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Step 4.1.1

Differentiate $2 x$ .

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

Step 4.1.2

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 4.1.3

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 4.1.4

Multiply $2$ by $1$ .

$2$

Step 4.2

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

$u_{lower} = 2 \cdot 0$

Step 4.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 4.4

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

$u_{upper} = 2 \frac{π}{4}$

Step 4.5

Cancel the common factor of $2$ .

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Step 4.5.1

Factor $2$ out of $4$ .

$u_{upper} = 2 \frac{π}{2 (2)}$

Step 4.5.2

Cancel the common factor.

$u_{upper} = 2 \frac{π}{2 \cdot 2}$

Step 4.5.3

Rewrite the expression.

$u_{upper} = \frac{π}{2}$

Step 4.6

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 4.7

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

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

Multiply $2$ by $2$ .

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

Step 8

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

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

Step 9

Substitute and simplify.

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

Simplify.

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Step 9.3.1

Combine $2$ and $\frac{π}{4}$ .

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

Step 9.3.2

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

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Step 9.3.2.1

Factor $2$ out of $2 π$ .

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

Step 9.3.2.2

Cancel the common factors.

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Step 9.3.2.2.1

Factor $2$ out of $4$ .

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

Step 9.3.2.2.2

Cancel the common factor.

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

Step 9.3.2.2.3

Rewrite the expression.

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

Step 9.3.3

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

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

Step 9.3.4

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

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

Step 9.3.5

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

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

Step 9.3.6

Multiply $4$ by $2$ .

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

Step 9.3.7

Multiply $2$ by $0$ .

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

Step 9.3.8

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

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

Step 9.3.9

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

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Step 9.3.9.1

Factor $2$ out of $0$ .

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

Step 9.3.9.2

Cancel the common factors.

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Step 9.3.9.2.1

Factor $2$ out of $2$ .

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

Step 9.3.9.2.2

Cancel the common factor.

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

Step 9.3.9.2.3

Rewrite the expression.

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

Step 9.3.9.2.4

Divide $0$ by $1$ .

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

Step 9.3.10

Multiply $- 1$ by $0$ .

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

Step 9.3.11

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

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

Step 9.3.12

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

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

Step 9.3.13

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

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

Step 9.3.14

Combine the numerators over the common denominator.

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

Step 9.3.15

Multiply $8$ by $- 1$ .

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

Step 9.3.16

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

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

Step 9.3.17

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

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Step 9.3.17.1

Factor $4$ out of $- 8$ .

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

Step 9.3.17.2

Cancel the common factors.

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Step 9.3.17.2.1

Factor $4$ out of $4$ .

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

Step 9.3.17.2.2

Cancel the common factor.

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

Step 9.3.17.2.3

Rewrite the expression.

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

Step 9.3.17.2.4

Divide $- 2$ by $1$ .

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

Step 10

Simplify.

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

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

$\frac{π \cdot 1 - 2 (- 0 + 1)}{8}$

Step 10.4

Multiply $π$ by $1$ .

$\frac{π - 2 (- 0 + 1)}{8}$

Step 10.5

Multiply $- 1$ by $0$ .

$\frac{π - 2 (0 + 1)}{8}$

Step 10.6

Add $0$ and $1$ .

$\frac{π - 2 \cdot 1}{8}$

Step 10.7

Multiply $- 2$ by $1$ .

$\frac{π - 2}{8}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{π - 2}{8}$

Decimal Form:

$0.14269908 \dots$