Calculus Examples

$\int_{\frac{π}{4}}^{\frac{π}{3}} x \cos (4 x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \cos (4 x)$ .

$x (\frac{1}{4} \sin (4 x))]_{\frac{π}{4}}^{\frac{π}{3}} - \int_{\frac{π}{4}}^{\frac{π}{3}} \frac{1}{4} \sin (4 x) d x$

Step 2

Simplify.

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Step 2.1

Combine $\frac{1}{4}$ and $\sin (4 x)$ .

$x \frac{\sin (4 x)}{4}]_{\frac{π}{4}}^{\frac{π}{3}} - \int_{\frac{π}{4}}^{\frac{π}{3}} \frac{1}{4} \sin (4 x) d x$

Step 2.2

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

$\frac{x \sin (4 x)}{4}]_{\frac{π}{4}}^{\frac{π}{3}} - \int_{\frac{π}{4}}^{\frac{π}{3}} \frac{1}{4} \sin (4 x) d x$

Step 3

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

$\frac{x \sin (4 x)}{4}]_{\frac{π}{4}}^{\frac{π}{3}} - (\frac{1}{4} \int_{\frac{π}{4}}^{\frac{π}{3}} \sin (4 x) d x)$

Step 4

Let $u = 4 x$ . Then $d u = 4 d x$ , so $\frac{1}{4} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = 4 x$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $4 x$ .

$\frac{d}{d x} [4 x]$

Step 4.1.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x]$

Step 4.1.3

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

$4 \cdot 1$

Step 4.1.4

Multiply $4$ by $1$ .

$4$

Step 4.2

Substitute the lower limit in for $x$ in $u = 4 x$ .

$u_{lower} = 4 \frac{π}{4}$

Step 4.3

Cancel the common factor of $4$ .

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Step 4.3.1

Cancel the common factor.

$u_{lower} = 4 \frac{π}{4}$

Step 4.3.2

Rewrite the expression.

$u_{lower} = π$

Step 4.4

Substitute the upper limit in for $x$ in $u = 4 x$ .

$u_{upper} = 4 \frac{π}{3}$

Step 4.5

Combine $4$ and $\frac{π}{3}$ .

$u_{upper} = \frac{4 π}{3}$

Step 4.6

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

$u_{lower} = π$

$u_{upper} = \frac{4 π}{3}$

Step 4.7

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

$\frac{x \sin (4 x)}{4}]_{\frac{π}{4}}^{\frac{π}{3}} - \frac{1}{4} \int_{π}^{\frac{4 π}{3}} \sin (u) \frac{1}{4} d u$

Step 5

Combine $\sin (u)$ and $\frac{1}{4}$ .

$\frac{x \sin (4 x)}{4}]_{\frac{π}{4}}^{\frac{π}{3}} - \frac{1}{4} \int_{π}^{\frac{4 π}{3}} \frac{\sin (u)}{4} d u$

Step 6

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$\frac{x \sin (4 x)}{4}]_{\frac{π}{4}}^{\frac{π}{3}} - \frac{1}{4} (\frac{1}{4} \int_{π}^{\frac{4 π}{3}} \sin (u) d u)$

Step 7

Simplify.

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Step 7.1

Multiply $\frac{1}{4}$ by $\frac{1}{4}$ .

$\frac{x \sin (4 x)}{4}]_{\frac{π}{4}}^{\frac{π}{3}} - \frac{1}{4 \cdot 4} \int_{π}^{\frac{4 π}{3}} \sin (u) d u$

Step 7.2

Multiply $4$ by $4$ .

$\frac{x \sin (4 x)}{4}]_{\frac{π}{4}}^{\frac{π}{3}} - \frac{1}{16} \int_{π}^{\frac{4 π}{3}} \sin (u) d u$

Step 8

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

$\frac{x \sin (4 x)}{4}]_{\frac{π}{4}}^{\frac{π}{3}} - \frac{1}{16} - \cos (u)]_{π}^{\frac{4 π}{3}}$

Step 9

Substitute and simplify.

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Step 9.1

Evaluate $\frac{x \sin (4 x)}{4}$ at $\frac{π}{3}$ and at $\frac{π}{4}$ .

$(\frac{\frac{π}{3} \sin (4 \frac{π}{3})}{4}) - \frac{\frac{π}{4} \sin (4 \frac{π}{4})}{4} - \frac{1}{16} - \cos (u)]_{π}^{\frac{4 π}{3}}$

Step 9.2

Evaluate $- \cos (u)$ at $\frac{4 π}{3}$ and at $π$ .

$(\frac{\frac{π}{3} \sin (4 \frac{π}{3})}{4}) - \frac{\frac{π}{4} \sin (4 \frac{π}{4})}{4} - \frac{1}{16} ((- \cos (\frac{4 π}{3})) + \cos (π))$

Step 9.3

Simplify.

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Step 9.3.1

Combine $4$ and $\frac{π}{3}$ .

$\frac{\frac{π}{3} \sin (\frac{4 π}{3})}{4} - \frac{\frac{π}{4} \sin (4 \frac{π}{4})}{4} - \frac{1}{16} ((- \cos (\frac{4 π}{3})) + \cos (π))$

Step 9.3.2

Combine $\frac{π}{3}$ and $\sin (\frac{4 π}{3})$ .

$\frac{\frac{π \sin (\frac{4 π}{3})}{3}}{4} - \frac{\frac{π}{4} \sin (4 \frac{π}{4})}{4} - \frac{1}{16} ((- \cos (\frac{4 π}{3})) + \cos (π))$

Step 9.3.3

Rewrite $\frac{\frac{π \sin (\frac{4 π}{3})}{3}}{4}$ as a product.

$\frac{π \sin (\frac{4 π}{3})}{3} \cdot \frac{1}{4} - \frac{\frac{π}{4} \sin (4 \frac{π}{4})}{4} - \frac{1}{16} ((- \cos (\frac{4 π}{3})) + \cos (π))$

Step 9.3.4

Multiply $\frac{π \sin (\frac{4 π}{3})}{3}$ by $\frac{1}{4}$ .

$\frac{π \sin (\frac{4 π}{3})}{3 \cdot 4} - \frac{\frac{π}{4} \sin (4 \frac{π}{4})}{4} - \frac{1}{16} ((- \cos (\frac{4 π}{3})) + \cos (π))$

Step 9.3.5

Multiply $3$ by $4$ .

$\frac{π \sin (\frac{4 π}{3})}{12} - \frac{\frac{π}{4} \sin (4 \frac{π}{4})}{4} - \frac{1}{16} ((- \cos (\frac{4 π}{3})) + \cos (π))$

Step 9.3.6

Combine $4$ and $\frac{π}{4}$ .

$\frac{π \sin (\frac{4 π}{3})}{12} - \frac{\frac{π}{4} \sin (\frac{4 π}{4})}{4} - \frac{1}{16} ((- \cos (\frac{4 π}{3})) + \cos (π))$

Step 9.3.7

Cancel the common factor of $4$ .

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Step 9.3.7.1

Cancel the common factor.

$\frac{π \sin (\frac{4 π}{3})}{12} - \frac{\frac{π}{4} \sin (\frac{4 π}{4})}{4} - \frac{1}{16} ((- \cos (\frac{4 π}{3})) + \cos (π))$

Step 9.3.7.2

Divide $π$ by $1$ .

$\frac{π \sin (\frac{4 π}{3})}{12} - \frac{\frac{π}{4} \sin (π)}{4} - \frac{1}{16} ((- \cos (\frac{4 π}{3})) + \cos (π))$

Step 9.3.8

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

$\frac{π \sin (\frac{4 π}{3})}{12} - \frac{\frac{π \sin (π)}{4}}{4} - \frac{1}{16} ((- \cos (\frac{4 π}{3})) + \cos (π))$

Step 9.3.9

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

$\frac{π \sin (\frac{4 π}{3})}{12} - (\frac{π \sin (π)}{4} \cdot \frac{1}{4}) - \frac{1}{16} ((- \cos (\frac{4 π}{3})) + \cos (π))$

Step 9.3.10

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

$\frac{π \sin (\frac{4 π}{3})}{12} - \frac{π \sin (π)}{4 \cdot 4} - \frac{1}{16} ((- \cos (\frac{4 π}{3})) + \cos (π))$

Step 9.3.11

Multiply $4$ by $4$ .

$\frac{π \sin (\frac{4 π}{3})}{12} - \frac{π \sin (π)}{16} - \frac{1}{16} ((- \cos (\frac{4 π}{3})) + \cos (π))$

Step 9.3.12

To write $- \frac{1}{16} (- \cos (\frac{4 π}{3}) + \cos (π))$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$\frac{π \sin (\frac{4 π}{3})}{12} - \frac{1}{16} (- \cos (\frac{4 π}{3}) + \cos (π)) \cdot \frac{12}{12} - \frac{π \sin (π)}{16}$

Step 9.3.13

Combine $- \frac{1}{16} (- \cos (\frac{4 π}{3}) + \cos (π))$ and $\frac{12}{12}$ .

$\frac{π \sin (\frac{4 π}{3})}{12} + \frac{- \frac{1}{16} (- \cos (\frac{4 π}{3}) + \cos (π)) \cdot 12}{12} - \frac{π \sin (π)}{16}$

Step 9.3.14

Combine the numerators over the common denominator.

$\frac{π \sin (\frac{4 π}{3}) - \frac{1}{16} (- \cos (\frac{4 π}{3}) + \cos (π)) \cdot 12}{12} - \frac{π \sin (π)}{16}$

Step 9.3.15

Multiply $12$ by $- 1$ .

$\frac{π \sin (\frac{4 π}{3}) - 12 (\frac{1}{16}) (- \cos (\frac{4 π}{3}) + \cos (π))}{12} - \frac{π \sin (π)}{16}$

Step 9.3.16

Combine $- 12$ and $\frac{1}{16}$ .

$\frac{π \sin (\frac{4 π}{3}) + \frac{- 12}{16} (- \cos (\frac{4 π}{3}) + \cos (π))}{12} - \frac{π \sin (π)}{16}$

Step 9.3.17

Cancel the common factor of $- 12$ and $16$ .

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Step 9.3.17.1

Factor $4$ out of $- 12$ .

$\frac{π \sin (\frac{4 π}{3}) + \frac{4 (- 3)}{16} (- \cos (\frac{4 π}{3}) + \cos (π))}{12} - \frac{π \sin (π)}{16}$

Step 9.3.17.2

Cancel the common factors.

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Step 9.3.17.2.1

Factor $4$ out of $16$ .

$\frac{π \sin (\frac{4 π}{3}) + \frac{4 \cdot - 3}{4 \cdot 4} (- \cos (\frac{4 π}{3}) + \cos (π))}{12} - \frac{π \sin (π)}{16}$

Step 9.3.17.2.2

Cancel the common factor.

$\frac{π \sin (\frac{4 π}{3}) + \frac{4 \cdot - 3}{4 \cdot 4} (- \cos (\frac{4 π}{3}) + \cos (π))}{12} - \frac{π \sin (π)}{16}$

Step 9.3.17.2.3

Rewrite the expression.

$\frac{π \sin (\frac{4 π}{3}) + \frac{- 3}{4} (- \cos (\frac{4 π}{3}) + \cos (π))}{12} - \frac{π \sin (π)}{16}$

Step 9.3.18

Move the negative in front of the fraction.

$\frac{π \sin (\frac{4 π}{3}) - \frac{3}{4} (- \cos (\frac{4 π}{3}) + \cos (π))}{12} - \frac{π \sin (π)}{16}$

Step 10

Simplify.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

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

$\frac{π (- \sin (\frac{π}{3})) - \frac{3}{4} (- \cos (\frac{4 π}{3}) + \cos (π))}{12} - \frac{π \sin (π)}{16}$

Step 10.1.2

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{π (- \frac{\sqrt{3}}{2}) - \frac{3}{4} (- \cos (\frac{4 π}{3}) + \cos (π))}{12} - \frac{π \sin (π)}{16}$

Step 10.1.3

Combine $π$ and $\frac{\sqrt{3}}{2}$ .

$\frac{- \frac{π \sqrt{3}}{2} - \frac{3}{4} (- \cos (\frac{4 π}{3}) + \cos (π))}{12} - \frac{π \sin (π)}{16}$

Step 10.1.4

Simplify each term.

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Step 10.1.4.1

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 third quadrant.

$\frac{- \frac{π \sqrt{3}}{2} - \frac{3}{4} (- - \cos (\frac{π}{3}) + \cos (π))}{12} - \frac{π \sin (π)}{16}$

Step 10.1.4.2

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$\frac{- \frac{π \sqrt{3}}{2} - \frac{3}{4} (- - \frac{1}{2} + \cos (π))}{12} - \frac{π \sin (π)}{16}$

Step 10.1.4.3

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

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Step 10.1.4.3.1

Multiply $- 1$ by $- 1$ .

$\frac{- \frac{π \sqrt{3}}{2} - \frac{3}{4} (1 (\frac{1}{2}) + \cos (π))}{12} - \frac{π \sin (π)}{16}$

Step 10.1.4.3.2

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

$\frac{- \frac{π \sqrt{3}}{2} - \frac{3}{4} (\frac{1}{2} + \cos (π))}{12} - \frac{π \sin (π)}{16}$

Step 10.1.4.4

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.

$\frac{- \frac{π \sqrt{3}}{2} - \frac{3}{4} (\frac{1}{2} - \cos (0))}{12} - \frac{π \sin (π)}{16}$

Step 10.1.4.5

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

$\frac{- \frac{π \sqrt{3}}{2} - \frac{3}{4} (\frac{1}{2} - 1 \cdot 1)}{12} - \frac{π \sin (π)}{16}$

Step 10.1.4.6

Multiply $- 1$ by $1$ .

$\frac{- \frac{π \sqrt{3}}{2} - \frac{3}{4} (\frac{1}{2} - 1)}{12} - \frac{π \sin (π)}{16}$

Step 10.1.5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{- \frac{π \sqrt{3}}{2} - \frac{3}{4} (\frac{1}{2} - 1 \cdot \frac{2}{2})}{12} - \frac{π \sin (π)}{16}$

Step 10.1.6

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

$\frac{- \frac{π \sqrt{3}}{2} - \frac{3}{4} (\frac{1}{2} + \frac{- 1 \cdot 2}{2})}{12} - \frac{π \sin (π)}{16}$

Step 10.1.7

Combine the numerators over the common denominator.

$\frac{- \frac{π \sqrt{3}}{2} - \frac{3}{4} \cdot \frac{1 - 1 \cdot 2}{2}}{12} - \frac{π \sin (π)}{16}$

Step 10.1.8

Simplify the numerator.

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Step 10.1.8.1

Multiply $- 1$ by $2$ .

$\frac{- \frac{π \sqrt{3}}{2} - \frac{3}{4} \cdot \frac{1 - 2}{2}}{12} - \frac{π \sin (π)}{16}$

Step 10.1.8.2

Subtract $2$ from $1$ .

$\frac{- \frac{π \sqrt{3}}{2} - \frac{3}{4} \cdot \frac{- 1}{2}}{12} - \frac{π \sin (π)}{16}$

Step 10.1.9

Move the negative in front of the fraction.

$\frac{- \frac{π \sqrt{3}}{2} - \frac{3}{4} (- \frac{1}{2})}{12} - \frac{π \sin (π)}{16}$

Step 10.1.10

Multiply $- \frac{3}{4} (- \frac{1}{2})$ .

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Step 10.1.10.1

Multiply $- 1$ by $- 1$ .

$\frac{- \frac{π \sqrt{3}}{2} + 1 (\frac{3}{4}) \frac{1}{2}}{12} - \frac{π \sin (π)}{16}$

Step 10.1.10.2

Multiply $\frac{3}{4}$ by $1$ .

$\frac{- \frac{π \sqrt{3}}{2} + \frac{3}{4} \cdot \frac{1}{2}}{12} - \frac{π \sin (π)}{16}$

Step 10.1.10.3

Multiply $\frac{3}{4}$ by $\frac{1}{2}$ .

$\frac{- \frac{π \sqrt{3}}{2} + \frac{3}{4 \cdot 2}}{12} - \frac{π \sin (π)}{16}$

Step 10.1.10.4

Multiply $4$ by $2$ .

$\frac{- \frac{π \sqrt{3}}{2} + \frac{3}{8}}{12} - \frac{π \sin (π)}{16}$

Step 10.1.11

To write $- \frac{π \sqrt{3}}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{- \frac{π \sqrt{3}}{2} \cdot \frac{4}{4} + \frac{3}{8}}{12} - \frac{π \sin (π)}{16}$

Step 10.1.12

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.1.12.1

Multiply $\frac{π \sqrt{3}}{2}$ by $\frac{4}{4}$ .

$\frac{- \frac{π \sqrt{3} \cdot 4}{2 \cdot 4} + \frac{3}{8}}{12} - \frac{π \sin (π)}{16}$

Step 10.1.12.2

Multiply $2$ by $4$ .

$\frac{- \frac{π \sqrt{3} \cdot 4}{8} + \frac{3}{8}}{12} - \frac{π \sin (π)}{16}$

Step 10.1.13

Combine the numerators over the common denominator.

$\frac{\frac{- π \sqrt{3} \cdot 4 + 3}{8}}{12} - \frac{π \sin (π)}{16}$

Step 10.1.14

Multiply $4$ by $- 1$ .

$\frac{\frac{- 4 π \sqrt{3} + 3}{8}}{12} - \frac{π \sin (π)}{16}$

Step 10.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{- 4 π \sqrt{3} + 3}{8} \cdot \frac{1}{12} - \frac{π \sin (π)}{16}$

Step 10.3

Multiply $\frac{- 4 π \sqrt{3} + 3}{8} \cdot \frac{1}{12}$ .

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Step 10.3.1

Multiply $\frac{- 4 π \sqrt{3} + 3}{8}$ by $\frac{1}{12}$ .

$\frac{- 4 π \sqrt{3} + 3}{8 \cdot 12} - \frac{π \sin (π)}{16}$

Step 10.3.2

Multiply $8$ by $12$ .

$\frac{- 4 π \sqrt{3} + 3}{96} - \frac{π \sin (π)}{16}$

Step 10.4

Factor $- 1$ out of $- 4 π \sqrt{3}$ .

$\frac{- (4 π \sqrt{3}) + 3}{96} - \frac{π \sin (π)}{16}$

Step 10.5

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{- (4 π \sqrt{3}) - 1 (- 3)}{96} - \frac{π \sin (π)}{16}$

Step 10.6

Factor $- 1$ out of $- (4 π \sqrt{3}) - 1 (- 3)$ .

$\frac{- (4 π \sqrt{3} - 3)}{96} - \frac{π \sin (π)}{16}$

Step 10.7

Rewrite $- (4 π \sqrt{3} - 3)$ as $- 1 (4 π \sqrt{3} - 3)$ .

$\frac{- 1 (4 π \sqrt{3} - 3)}{96} - \frac{π \sin (π)}{16}$

Step 10.8

Move the negative in front of the fraction.

$- \frac{4 π \sqrt{3} - 3}{96} - \frac{π \sin (π)}{16}$

Step 10.9

Simplify the numerator.

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Step 10.9.1

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

$- \frac{4 π \sqrt{3} - 3}{96} - \frac{π \sin (0)}{16}$

Step 10.9.2

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

$- \frac{4 π \sqrt{3} - 3}{96} - \frac{π \cdot 0}{16}$

Step 10.10

Multiply $π$ by $0$ .

$- \frac{4 π \sqrt{3} - 3}{96} - \frac{0}{16}$

Step 10.11

Divide $0$ by $16$ .

$- \frac{4 π \sqrt{3} - 3}{96} - 0$

Step 11

Subtract $0$ from $- \frac{4 π \sqrt{3} - 3}{96}$ .

$- \frac{4 π \sqrt{3} - 3}{96}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$- \frac{4 π \sqrt{3} - 3}{96}$

Decimal Form:

$- 0.19547492 \dots$