Calculus Examples

$\int_{0}^{π} x \cos (3 x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \cos (3 x)$ .

$x (\frac{1}{3} \sin (3 x))]_{0}^{π} - \int_{0}^{π} \frac{1}{3} \sin (3 x) d x$

Step 2

Simplify.

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Step 2.1

Combine $\frac{1}{3}$ and $\sin (3 x)$ .

$x \frac{\sin (3 x)}{3}]_{0}^{π} - \int_{0}^{π} \frac{1}{3} \sin (3 x) d x$

Step 2.2

Combine $x$ and $\frac{\sin (3 x)}{3}$ .

$\frac{x \sin (3 x)}{3}]_{0}^{π} - \int_{0}^{π} \frac{1}{3} \sin (3 x) d x$

Step 3

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

$\frac{x \sin (3 x)}{3}]_{0}^{π} - (\frac{1}{3} \int_{0}^{π} \sin (3 x) d x)$

Step 4

Let $u = 3 x$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = 3 x$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $3 x$ .

$\frac{d}{d x} [3 x]$

Step 4.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x]$

Step 4.1.3

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

$3 \cdot 1$

Step 4.1.4

Multiply $3$ by $1$ .

$3$

Step 4.2

Substitute the lower limit in for $x$ in $u = 3 x$ .

$u_{lower} = 3 \cdot 0$

Step 4.3

Multiply $3$ by $0$ .

$u_{lower} = 0$

Step 4.4

Substitute the upper limit in for $x$ in $u = 3 x$ .

$u_{upper} = 3 π$

Step 4.5

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

$u_{lower} = 0$

$u_{upper} = 3 π$

Step 4.6

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

$\frac{x \sin (3 x)}{3}]_{0}^{π} - \frac{1}{3} \int_{0}^{3 π} \sin (u) \frac{1}{3} d u$

Step 5

Combine $\sin (u)$ and $\frac{1}{3}$ .

$\frac{x \sin (3 x)}{3}]_{0}^{π} - \frac{1}{3} \int_{0}^{3 π} \frac{\sin (u)}{3} d u$

Step 6

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$\frac{x \sin (3 x)}{3}]_{0}^{π} - \frac{1}{3} (\frac{1}{3} \int_{0}^{3 π} \sin (u) d u)$

Step 7

Simplify.

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Step 7.1

Multiply $\frac{1}{3}$ by $\frac{1}{3}$ .

$\frac{x \sin (3 x)}{3}]_{0}^{π} - \frac{1}{3 \cdot 3} \int_{0}^{3 π} \sin (u) d u$

Step 7.2

Multiply $3$ by $3$ .

$\frac{x \sin (3 x)}{3}]_{0}^{π} - \frac{1}{9} \int_{0}^{3 π} \sin (u) d u$

Step 8

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

$\frac{x \sin (3 x)}{3}]_{0}^{π} - \frac{1}{9} - \cos (u)]_{0}^{3 π}$

Step 9

Substitute and simplify.

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Step 9.1

Evaluate $\frac{x \sin (3 x)}{3}$ at $π$ and at $0$ .

$(\frac{π \sin (3 π)}{3}) - \frac{0 \sin (3 \cdot 0)}{3} - \frac{1}{9} - \cos (u)]_{0}^{3 π}$

Step 9.2

Evaluate $- \cos (u)$ at $3 π$ and at $0$ .

$(\frac{π \sin (3 π)}{3}) - \frac{0 \sin (3 \cdot 0)}{3} - \frac{1}{9} ((- \cos (3 π)) + \cos (0))$

Step 9.3

Simplify.

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Step 9.3.1

Multiply $3$ by $0$ .

$\frac{π \sin (3 π)}{3} - \frac{0 \sin (0)}{3} - \frac{1}{9} ((- \cos (3 π)) + \cos (0))$

Step 9.3.2

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

$\frac{π \sin (3 π)}{3} - \frac{0}{3} - \frac{1}{9} ((- \cos (3 π)) + \cos (0))$

Step 9.3.3

Cancel the common factor of $0$ and $3$ .

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Step 9.3.3.1

Factor $3$ out of $0$ .

$\frac{π \sin (3 π)}{3} - \frac{3 (0)}{3} - \frac{1}{9} ((- \cos (3 π)) + \cos (0))$

Step 9.3.3.2

Cancel the common factors.

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Step 9.3.3.2.1

Factor $3$ out of $3$ .

$\frac{π \sin (3 π)}{3} - \frac{3 \cdot 0}{3 \cdot 1} - \frac{1}{9} ((- \cos (3 π)) + \cos (0))$

Step 9.3.3.2.2

Cancel the common factor.

$\frac{π \sin (3 π)}{3} - \frac{3 \cdot 0}{3 \cdot 1} - \frac{1}{9} ((- \cos (3 π)) + \cos (0))$

Step 9.3.3.2.3

Rewrite the expression.

$\frac{π \sin (3 π)}{3} - \frac{0}{1} - \frac{1}{9} ((- \cos (3 π)) + \cos (0))$

Step 9.3.3.2.4

Divide $0$ by $1$ .

$\frac{π \sin (3 π)}{3} - 0 - \frac{1}{9} ((- \cos (3 π)) + \cos (0))$

Step 9.3.4

Multiply $- 1$ by $0$ .

$\frac{π \sin (3 π)}{3} + 0 - \frac{1}{9} ((- \cos (3 π)) + \cos (0))$

Step 9.3.5

Add $\frac{π \sin (3 π)}{3}$ and $0$ .

$\frac{π \sin (3 π)}{3} - \frac{1}{9} ((- \cos (3 π)) + \cos (0))$

Step 9.3.6

To write $- \frac{1}{9} (- \cos (3 π) + \cos (0))$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{π \sin (3 π)}{3} - \frac{1}{9} (- \cos (3 π) + \cos (0)) \cdot \frac{3}{3}$

Step 9.3.7

Combine $- \frac{1}{9} (- \cos (3 π) + \cos (0))$ and $\frac{3}{3}$ .

$\frac{π \sin (3 π)}{3} + \frac{- \frac{1}{9} (- \cos (3 π) + \cos (0)) \cdot 3}{3}$

Step 9.3.8

Combine the numerators over the common denominator.

$\frac{π \sin (3 π) - \frac{1}{9} (- \cos (3 π) + \cos (0)) \cdot 3}{3}$

Step 9.3.9

Multiply $3$ by $- 1$ .

$\frac{π \sin (3 π) - 3 (\frac{1}{9}) (- \cos (3 π) + \cos (0))}{3}$

Step 9.3.10

Combine $- 3$ and $\frac{1}{9}$ .

$\frac{π \sin (3 π) + \frac{- 3}{9} (- \cos (3 π) + \cos (0))}{3}$

Step 9.3.11

Cancel the common factor of $- 3$ and $9$ .

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Step 9.3.11.1

Factor $3$ out of $- 3$ .

$\frac{π \sin (3 π) + \frac{3 (- 1)}{9} (- \cos (3 π) + \cos (0))}{3}$

Step 9.3.11.2

Cancel the common factors.

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Step 9.3.11.2.1

Factor $3$ out of $9$ .

$\frac{π \sin (3 π) + \frac{3 \cdot - 1}{3 \cdot 3} (- \cos (3 π) + \cos (0))}{3}$

Step 9.3.11.2.2

Cancel the common factor.

$\frac{π \sin (3 π) + \frac{3 \cdot - 1}{3 \cdot 3} (- \cos (3 π) + \cos (0))}{3}$

Step 9.3.11.2.3

Rewrite the expression.

$\frac{π \sin (3 π) + \frac{- 1}{3} (- \cos (3 π) + \cos (0))}{3}$

Step 9.3.12

Move the negative in front of the fraction.

$\frac{π \sin (3 π) - \frac{1}{3} (- \cos (3 π) + \cos (0))}{3}$

Step 10

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

$\frac{π \sin (3 π) - \frac{1}{3} (- \cos (3 π) + 1)}{3}$

Step 11

Simplify.

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Step 11.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{π \sin (π) - \frac{1}{3} (- \cos (3 π) + 1)}{3}$

Step 11.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{π \sin (0) - \frac{1}{3} (- \cos (3 π) + 1)}{3}$

Step 11.3

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

$\frac{π \cdot 0 - \frac{1}{3} (- \cos (3 π) + 1)}{3}$

Step 11.4

Multiply $π$ by $0$ .

$\frac{0 - \frac{1}{3} (- \cos (3 π) + 1)}{3}$

Step 11.5

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{0 - \frac{1}{3} (- \cos (π) + 1)}{3}$

Step 11.6

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$\frac{0 - \frac{1}{3} (- - \cos (0) + 1)}{3}$

Step 11.7

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

$\frac{0 - \frac{1}{3} (- (- 1 \cdot 1) + 1)}{3}$

Step 11.8

Multiply $- 1$ by $1$ .

$\frac{0 - \frac{1}{3} (- - 1 + 1)}{3}$

Step 11.9

Multiply $- 1$ by $- 1$ .

$\frac{0 - \frac{1}{3} (1 + 1)}{3}$

Step 11.10

Add $1$ and $1$ .

$\frac{0 - \frac{1}{3} \cdot 2}{3}$

Step 11.11

Multiply $- \frac{1}{3} \cdot 2$ .

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Step 11.11.1

Multiply $2$ by $- 1$ .

$\frac{0 - 2 (\frac{1}{3})}{3}$

Step 11.11.2

Combine $- 2$ and $\frac{1}{3}$ .

$\frac{0 + \frac{- 2}{3}}{3}$

Step 11.12

Move the negative in front of the fraction.

$\frac{0 - \frac{2}{3}}{3}$

Step 11.13

Subtract $\frac{2}{3}$ from $0$ .

$\frac{- \frac{2}{3}}{3}$

Step 11.14

Multiply the numerator by the reciprocal of the denominator.

$- \frac{2}{3} \cdot \frac{1}{3}$

Step 11.15

Multiply $- \frac{2}{3} \cdot \frac{1}{3}$ .

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Step 11.15.1

Multiply $\frac{1}{3}$ by $\frac{2}{3}$ .

$- \frac{2}{3 \cdot 3}$

Step 11.15.2

Multiply $3$ by $3$ .

$- \frac{2}{9}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$- \frac{2}{9}$

Decimal Form:

$- 0.\overline{2}$