Calculus Examples

$y = \tan (5 x)$ , $y = 2 \sin (5 x)$ , $- \frac{π}{15} \leq x \leq \frac{π}{15}$

Step 1

Solve by substitution to find the intersection between the curves.

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Step 1.1

Eliminate the equal sides of each equation and combine.

$\tan (5 x) = 2 \sin (5 x)$

Step 1.2

Solve $\tan (5 x) = 2 \sin (5 x)$ for $x$ .

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Step 1.2.1

Divide each term in $\tan (5 x) = 2 \sin (5 x)$ by $\tan (5 x)$ and simplify.

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Step 1.2.1.1

Divide each term in $\tan (5 x) = 2 \sin (5 x)$ by $\tan (5 x)$ .

$\frac{\tan (5 x)}{\tan (5 x)} = \frac{2 \sin (5 x)}{\tan (5 x)}$

Step 1.2.1.2

Simplify the left side.

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Step 1.2.1.2.1

Cancel the common factor of $\tan (5 x)$ .

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Step 1.2.1.2.1.1

Cancel the common factor.

$\frac{\tan (5 x)}{\tan (5 x)} = \frac{2 \sin (5 x)}{\tan (5 x)}$

Step 1.2.1.2.1.2

Rewrite the expression.

$1 = \frac{2 \sin (5 x)}{\tan (5 x)}$

Step 1.2.1.3

Simplify the right side.

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Step 1.2.1.3.1

Separate fractions.

$1 = \frac{2}{1} \cdot \frac{\sin (5 x)}{\tan (5 x)}$

Step 1.2.1.3.2

Rewrite $\tan (5 x)$ in terms of sines and cosines.

$1 = \frac{2}{1} \cdot \frac{\sin (5 x)}{\frac{\sin (5 x)}{\cos (5 x)}}$

Step 1.2.1.3.3

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (5 x)}{\cos (5 x)}$ .

$1 = \frac{2}{1} (\sin (5 x) \frac{\cos (5 x)}{\sin (5 x)})$

Step 1.2.1.3.4

Write $\sin (5 x)$ as a fraction with denominator $1$ .

$1 = \frac{2}{1} (\frac{\sin (5 x)}{1} \cdot \frac{\cos (5 x)}{\sin (5 x)})$

Step 1.2.1.3.5

Cancel the common factor of $\sin (5 x)$ .

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Step 1.2.1.3.5.1

Cancel the common factor.

$1 = \frac{2}{1} (\frac{\sin (5 x)}{1} \cdot \frac{\cos (5 x)}{\sin (5 x)})$

Step 1.2.1.3.5.2

Rewrite the expression.

$1 = \frac{2}{1} \cos (5 x)$

Step 1.2.1.3.6

Divide $2$ by $1$ .

$1 = 2 \cos (5 x)$

Step 1.2.2

Rewrite the equation as $2 \cos (5 x) = 1$ .

$2 \cos (5 x) = 1$

Step 1.2.3

Divide each term in $2 \cos (5 x) = 1$ by $2$ and simplify.

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Step 1.2.3.1

Divide each term in $2 \cos (5 x) = 1$ by $2$ .

$\frac{2 \cos (5 x)}{2} = \frac{1}{2}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $2$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{2 \cos (5 x)}{2} = \frac{1}{2}$

Step 1.2.3.2.1.2

Divide $\cos (5 x)$ by $1$ .

$\cos (5 x) = \frac{1}{2}$

Step 1.2.4

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$5 x = \arccos (\frac{1}{2})$

Step 1.2.5

Simplify the right side.

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Step 1.2.5.1

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

$5 x = \frac{π}{3}$

Step 1.2.6

Divide each term in $5 x = \frac{π}{3}$ by $5$ and simplify.

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Step 1.2.6.1

Divide each term in $5 x = \frac{π}{3}$ by $5$ .

$\frac{5 x}{5} = \frac{\frac{π}{3}}{5}$

Step 1.2.6.2

Simplify the left side.

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Step 1.2.6.2.1

Cancel the common factor of $5$ .

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Step 1.2.6.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{\frac{π}{3}}{5}$

Step 1.2.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{3}}{5}$

Step 1.2.6.3

Simplify the right side.

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Step 1.2.6.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{3} \cdot \frac{1}{5}$

Step 1.2.6.3.2

Multiply $\frac{π}{3} \cdot \frac{1}{5}$ .

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Step 1.2.6.3.2.1

Multiply $\frac{π}{3}$ by $\frac{1}{5}$ .

$x = \frac{π}{3 \cdot 5}$

Step 1.2.6.3.2.2

Multiply $3$ by $5$ .

$x = \frac{π}{15}$

Step 1.2.7

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$5 x = 2 π - \frac{π}{3}$

Step 1.2.8

Solve for $x$ .

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Step 1.2.8.1

Simplify.

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Step 1.2.8.1.1

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

$5 x = 2 π \cdot \frac{3}{3} - \frac{π}{3}$

Step 1.2.8.1.2

Combine $2 π$ and $\frac{3}{3}$ .

$5 x = \frac{2 π \cdot 3}{3} - \frac{π}{3}$

Step 1.2.8.1.3

Combine the numerators over the common denominator.

$5 x = \frac{2 π \cdot 3 - π}{3}$

Step 1.2.8.1.4

Multiply $3$ by $2$ .

$5 x = \frac{6 π - π}{3}$

Step 1.2.8.1.5

Subtract $π$ from $6 π$ .

$5 x = \frac{5 π}{3}$

Step 1.2.8.2

Divide each term in $5 x = \frac{5 π}{3}$ by $5$ and simplify.

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Step 1.2.8.2.1

Divide each term in $5 x = \frac{5 π}{3}$ by $5$ .

$\frac{5 x}{5} = \frac{\frac{5 π}{3}}{5}$

Step 1.2.8.2.2

Simplify the left side.

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Step 1.2.8.2.2.1

Cancel the common factor of $5$ .

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Step 1.2.8.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{\frac{5 π}{3}}{5}$

Step 1.2.8.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{5 π}{3}}{5}$

Step 1.2.8.2.3

Simplify the right side.

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Step 1.2.8.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{5 π}{3} \cdot \frac{1}{5}$

Step 1.2.8.2.3.2

Cancel the common factor of $5$ .

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Step 1.2.8.2.3.2.1

Factor $5$ out of $5 π$ .

$x = \frac{5 (π)}{3} \cdot \frac{1}{5}$

Step 1.2.8.2.3.2.2

Cancel the common factor.

$x = \frac{5 π}{3} \cdot \frac{1}{5}$

Step 1.2.8.2.3.2.3

Rewrite the expression.

$x = \frac{π}{3}$

Step 1.2.9

Find the period of $\cos (5 x)$ .

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Step 1.2.9.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 1.2.9.2

Replace $b$ with $5$ in the formula for period.

$\frac{2 π}{| 5 |}$

Step 1.2.9.3

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$\frac{2 π}{5}$

Step 1.2.10

The period of the $\cos (5 x)$ function is $\frac{2 π}{5}$ so values will repeat every $\frac{2 π}{5}$ radians in both directions.

$x = \frac{π}{15} + \frac{2 π n}{5}, \frac{π}{3} + \frac{2 π n}{5}$ , for any integer $n$

Step 1.3

Evaluate $y$ when $x = \frac{π}{15} + \frac{2 π n}{5}$ .

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Step 1.3.1

Substitute $\frac{π}{15} + \frac{2 π n}{5}$ for $x$ .

$y = 2 \sin (5 (\frac{π}{15} + \frac{2 π n}{5}))$

Step 1.3.2

Substitute $\frac{π}{15} + \frac{2 π n}{5}$ for $x$ in $y = 2 \sin (5 (\frac{π}{15} + \frac{2 π n}{5}))$ and solve for $y$ .

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Step 1.3.2.1

Remove parentheses.

$y = 2 \sin (5 (\frac{π}{15} + \frac{2 π n}{5}))$

Step 1.3.2.2

Simplify $2 \sin (5 (\frac{π}{15} + \frac{2 π n}{5}))$ .

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Step 1.3.2.2.1

Apply the distributive property.

$y = 2 \sin (5 \frac{π}{15} + 5 \frac{2 π n}{5})$

Step 1.3.2.2.2

Cancel the common factor of $5$ .

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Step 1.3.2.2.2.1

Factor $5$ out of $15$ .

$y = 2 \sin (5 \frac{π}{5 (3)} + 5 \frac{2 π n}{5})$

Step 1.3.2.2.2.2

Cancel the common factor.

$y = 2 \sin (5 \frac{π}{5 \cdot 3} + 5 \frac{2 π n}{5})$

Step 1.3.2.2.2.3

Rewrite the expression.

$y = 2 \sin (\frac{π}{3} + 5 \frac{2 π n}{5})$

Step 1.3.2.2.3

Cancel the common factor of $5$ .

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Step 1.3.2.2.3.1

Cancel the common factor.

$y = 2 \sin (\frac{π}{3} + 5 \frac{2 π n}{5})$

Step 1.3.2.2.3.2

Rewrite the expression.

$y = 2 \sin (\frac{π}{3} + 2 π n)$

Step 1.4

Evaluate $y$ when $x = \frac{π}{3} + \frac{2 π n}{5}$ .

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Step 1.4.1

Substitute $\frac{π}{3} + \frac{2 π n}{5}$ for $x$ .

$y = 2 \sin (5 (\frac{π}{3} + \frac{2 π n}{5}))$

Step 1.4.2

Simplify $2 \sin (5 (\frac{π}{3} + \frac{2 π n}{5}))$ .

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Step 1.4.2.1

Apply the distributive property.

$y = 2 \sin (5 \frac{π}{3} + 5 \frac{2 π n}{5})$

Step 1.4.2.2

Combine $5$ and $\frac{π}{3}$ .

$y = 2 \sin (\frac{5 π}{3} + 5 \frac{2 π n}{5})$

Step 1.4.2.3

Cancel the common factor of $5$ .

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Step 1.4.2.3.1

Cancel the common factor.

$y = 2 \sin (\frac{5 π}{3} + 5 \frac{2 π n}{5})$

Step 1.4.2.3.2

Rewrite the expression.

$y = 2 \sin (\frac{5 π}{3} + 2 π n)$

Step 1.5

List all of the solutions.

$y = 2 \sin (\frac{π}{3} + 2 π n), x = \frac{π}{15} + \frac{2 π n}{5}$

$y = 2 \sin (\frac{5 π}{3} + 2 π n), x = \frac{π}{3} + \frac{2 π n}{5}$

$y = 2 \sin (\frac{π}{3} + 2 π n), x = \frac{π}{15} + \frac{2 π n}{5}$

$y = 2 \sin (\frac{5 π}{3} + 2 π n), x = \frac{π}{3} + \frac{2 π n}{5}$

Step 2

The area of the region between the curves is defined as the integral of the upper curve minus the integral of the lower curve over each region. The regions are determined by the intersection points of the curves. This can be done algebraically or graphically.

$A r e a = \int_{- \frac{π}{15}}^{\frac{π}{15}} 2 \sin (5 x) d x - \int_{- \frac{π}{15}}^{\frac{π}{15}} \tan (5 x) d x$

Step 3

Integrate to find the area between $- \frac{π}{15}$ and $\frac{π}{15}$ .

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Step 3.1

Combine the integrals into a single integral.

$\int_{- \frac{π}{15}}^{\frac{π}{15}} 2 \sin (5 x) - (\tan (5 x)) d x$

Step 3.2

Rewrite $\tan (5 x)$ in terms of sines and cosines.

$\int_{- \frac{π}{15}}^{\frac{π}{15}} 2 \sin (5 x) - \frac{\sin (5 x)}{\cos (5 x)} d x$

Step 3.3

Convert from $\frac{\sin (5 x)}{\cos (5 x)}$ to $\tan (5 x)$ .

$\int_{- \frac{π}{15}}^{\frac{π}{15}} 2 \sin (5 x) - \tan (5 x) d x$

Step 3.4

Split the single integral into multiple integrals.

$\int_{- \frac{π}{15}}^{\frac{π}{15}} 2 \sin (5 x) d x + \int_{- \frac{π}{15}}^{\frac{π}{15}} - \tan (5 x) d x$

Step 3.5

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

$2 \int_{- \frac{π}{15}}^{\frac{π}{15}} \sin (5 x) d x + \int_{- \frac{π}{15}}^{\frac{π}{15}} - \tan (5 x) d x$

Step 3.6

Let $u_{1} = 5 x$ . Then $d u_{1} = 5 d x$ , so $\frac{1}{5} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 3.6.1

Let $u_{1} = 5 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 3.6.1.1

Differentiate $5 x$ .

$\frac{d}{d x} [5 x]$

Step 3.6.1.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$5 \frac{d}{d x} [x]$

Step 3.6.1.3

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

$5 \cdot 1$

Step 3.6.1.4

Multiply $5$ by $1$ .

$5$

Step 3.6.2

Substitute the lower limit in for $x$ in $u_{1} = 5 x$ .

$u_{lower} = 5 (- \frac{π}{15})$

Step 3.6.3

Simplify.

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Step 3.6.3.1

Cancel the common factor of $5$ .

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Step 3.6.3.1.1

Move the leading negative in $- \frac{π}{15}$ into the numerator.

$u_{lower} = 5 \frac{- π}{15}$

Step 3.6.3.1.2

Factor $5$ out of $15$ .

$u_{lower} = 5 \frac{- π}{5 (3)}$

Step 3.6.3.1.3

Cancel the common factor.

$u_{lower} = 5 \frac{- π}{5 \cdot 3}$

Step 3.6.3.1.4

Rewrite the expression.

$u_{lower} = \frac{- π}{3}$

Step 3.6.3.2

Move the negative in front of the fraction.

$u_{lower} = - \frac{π}{3}$

Step 3.6.4

Substitute the upper limit in for $x$ in $u_{1} = 5 x$ .

$u_{upper} = 5 \frac{π}{15}$

Step 3.6.5

Cancel the common factor of $5$ .

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Step 3.6.5.1

Factor $5$ out of $15$ .

$u_{upper} = 5 \frac{π}{5 (3)}$

Step 3.6.5.2

Cancel the common factor.

$u_{upper} = 5 \frac{π}{5 \cdot 3}$

Step 3.6.5.3

Rewrite the expression.

$u_{upper} = \frac{π}{3}$

Step 3.6.6

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

$u_{lower} = - \frac{π}{3}$

$u_{upper} = \frac{π}{3}$

Step 3.6.7

Rewrite the problem using $u_{1}$ , $d u_{1}$ , and the new limits of integration.

$2 \int_{- \frac{π}{3}}^{\frac{π}{3}} \sin (u_{1}) \frac{1}{5} d u_{1} + \int_{- \frac{π}{15}}^{\frac{π}{15}} - \tan (5 x) d x$

Step 3.7

Combine $\sin (u_{1})$ and $\frac{1}{5}$ .

$2 \int_{- \frac{π}{3}}^{\frac{π}{3}} \frac{\sin (u_{1})}{5} d u_{1} + \int_{- \frac{π}{15}}^{\frac{π}{15}} - \tan (5 x) d x$

Step 3.8

Since $\frac{1}{5}$ is constant with respect to $u_{1}$ , move $\frac{1}{5}$ out of the integral.

$2 (\frac{1}{5} \int_{- \frac{π}{3}}^{\frac{π}{3}} \sin (u_{1}) d u_{1}) + \int_{- \frac{π}{15}}^{\frac{π}{15}} - \tan (5 x) d x$

Step 3.9

Combine $\frac{1}{5}$ and $2$ .

$\frac{2}{5} \int_{- \frac{π}{3}}^{\frac{π}{3}} \sin (u_{1}) d u_{1} + \int_{- \frac{π}{15}}^{\frac{π}{15}} - \tan (5 x) d x$

Step 3.10

The integral of $\sin (u_{1})$ with respect to $u_{1}$ is $- \cos (u_{1})$ .

$\frac{2}{5} - \cos (u_{1})]_{- \frac{π}{3}}^{\frac{π}{3}} + \int_{- \frac{π}{15}}^{\frac{π}{15}} - \tan (5 x) d x$

Step 3.11

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{2}{5} - \cos (u_{1})]_{- \frac{π}{3}}^{\frac{π}{3}} - \int_{- \frac{π}{15}}^{\frac{π}{15}} \tan (5 x) d x$

Step 3.12

Let $u_{2} = 5 x$ . Then $d u_{2} = 5 d x$ , so $\frac{1}{5} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 3.12.1

Let $u_{2} = 5 x$ . Find $\frac{d u_{2}}{d x}$ .

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Step 3.12.1.1

Differentiate $5 x$ .

$\frac{d}{d x} [5 x]$

Step 3.12.1.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$5 \frac{d}{d x} [x]$

Step 3.12.1.3

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

$5 \cdot 1$

Step 3.12.1.4

Multiply $5$ by $1$ .

$5$

Step 3.12.2

Substitute the lower limit in for $x$ in $u_{2} = 5 x$ .

$u_{lower} = 5 (- \frac{π}{15})$

Step 3.12.3

Simplify.

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Step 3.12.3.1

Cancel the common factor of $5$ .

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Step 3.12.3.1.1

Move the leading negative in $- \frac{π}{15}$ into the numerator.

$u_{lower} = 5 \frac{- π}{15}$

Step 3.12.3.1.2

Factor $5$ out of $15$ .

$u_{lower} = 5 \frac{- π}{5 (3)}$

Step 3.12.3.1.3

Cancel the common factor.

$u_{lower} = 5 \frac{- π}{5 \cdot 3}$

Step 3.12.3.1.4

Rewrite the expression.

$u_{lower} = \frac{- π}{3}$

Step 3.12.3.2

Move the negative in front of the fraction.

$u_{lower} = - \frac{π}{3}$

Step 3.12.4

Substitute the upper limit in for $x$ in $u_{2} = 5 x$ .

$u_{upper} = 5 \frac{π}{15}$

Step 3.12.5

Cancel the common factor of $5$ .

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Step 3.12.5.1

Factor $5$ out of $15$ .

$u_{upper} = 5 \frac{π}{5 (3)}$

Step 3.12.5.2

Cancel the common factor.

$u_{upper} = 5 \frac{π}{5 \cdot 3}$

Step 3.12.5.3

Rewrite the expression.

$u_{upper} = \frac{π}{3}$

Step 3.12.6

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

$u_{lower} = - \frac{π}{3}$

$u_{upper} = \frac{π}{3}$

Step 3.12.7

Rewrite the problem using $u_{2}$ , $d u_{2}$ , and the new limits of integration.

$\frac{2}{5} - \cos (u_{1})]_{- \frac{π}{3}}^{\frac{π}{3}} - \int_{- \frac{π}{3}}^{\frac{π}{3}} \tan (u_{2}) \frac{1}{5} d u_{2}$

Step 3.13

Combine $\tan (u_{2})$ and $\frac{1}{5}$ .

$\frac{2}{5} - \cos (u_{1})]_{- \frac{π}{3}}^{\frac{π}{3}} - \int_{- \frac{π}{3}}^{\frac{π}{3}} \frac{\tan (u_{2})}{5} d u_{2}$

Step 3.14

Since $\frac{1}{5}$ is constant with respect to $u_{2}$ , move $\frac{1}{5}$ out of the integral.

$\frac{2}{5} - \cos (u_{1})]_{- \frac{π}{3}}^{\frac{π}{3}} - (\frac{1}{5} \int_{- \frac{π}{3}}^{\frac{π}{3}} \tan (u_{2}) d u_{2})$

Step 3.15

The integral of $\tan (u_{2})$ with respect to $u_{2}$ is $\ln (| \sec (u_{2}) |)$ .

$\frac{2}{5} - \cos (u_{1})]_{- \frac{π}{3}}^{\frac{π}{3}} - \frac{1}{5} \ln (| \sec (u_{2}) |)]_{- \frac{π}{3}}^{\frac{π}{3}}$

Step 3.16

Substitute and simplify.

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Step 3.16.1

Evaluate $- \cos (u_{1})$ at $\frac{π}{3}$ and at $- \frac{π}{3}$ .

$\frac{2}{5} ((- \cos (\frac{π}{3})) + \cos (- \frac{π}{3})) - \frac{1}{5} \ln (| \sec (u_{2}) |)]_{- \frac{π}{3}}^{\frac{π}{3}}$

Step 3.16.2

Evaluate $\ln (| \sec (u_{2}) |)$ at $\frac{π}{3}$ and at $- \frac{π}{3}$ .

$\frac{2}{5} ((- \cos (\frac{π}{3})) + \cos (- \frac{π}{3})) - \frac{1}{5} ((\ln (| \sec (\frac{π}{3}) |)) - \ln (| \sec (- \frac{π}{3}) |))$

Step 3.16.3

Remove parentheses.

$\frac{2}{5} (- \cos (\frac{π}{3}) + \cos (- \frac{π}{3})) - \frac{1}{5} (\ln (| \sec (\frac{π}{3}) |) - \ln (| \sec (- \frac{π}{3}) |))$

Step 3.17

Simplify.

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Step 3.17.1

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

$\frac{2}{5} (- \frac{1}{2} + \cos (- \frac{π}{3})) - \frac{1}{5} (\ln (| \sec (\frac{π}{3}) |) - \ln (| \sec (- \frac{π}{3}) |))$

Step 3.17.2

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

$\frac{2}{5} (- \frac{1}{2} + \cos (- \frac{π}{3})) - \frac{1}{5} (\ln (| 2 |) - \ln (| \sec (- \frac{π}{3}) |))$

Step 3.17.3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\frac{2}{5} (- \frac{1}{2} + \cos (- \frac{π}{3})) - \frac{1}{5} \ln (\frac{| 2 |}{| \sec (- \frac{π}{3}) |})$

Step 3.17.4

Combine $\ln (\frac{| 2 |}{| \sec (- \frac{π}{3}) |})$ and $\frac{1}{5}$ .

$\frac{2}{5} (- \frac{1}{2} + \cos (- \frac{π}{3})) - \frac{\ln (\frac{| 2 |}{| \sec (- \frac{π}{3}) |})}{5}$

Step 3.17.5

To write $\frac{2}{5} (- \frac{1}{2} + \cos (- \frac{π}{3}))$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{2}{5} (- \frac{1}{2} + \cos (- \frac{π}{3})) \cdot \frac{5}{5} - \frac{\ln (\frac{| 2 |}{| \sec (- \frac{π}{3}) |})}{5}$

Step 3.17.6

Combine $\frac{2}{5} (- \frac{1}{2} + \cos (- \frac{π}{3}))$ and $\frac{5}{5}$ .

$\frac{\frac{2}{5} (- \frac{1}{2} + \cos (- \frac{π}{3})) \cdot 5}{5} - \frac{\ln (\frac{| 2 |}{| \sec (- \frac{π}{3}) |})}{5}$

Step 3.17.7

Combine the numerators over the common denominator.

$\frac{\frac{2}{5} (- \frac{1}{2} + \cos (- \frac{π}{3})) \cdot 5 - \ln (\frac{| 2 |}{| \sec (- \frac{π}{3}) |})}{5}$

Step 3.17.8

Combine $5$ and $\frac{2}{5}$ .

$\frac{\frac{5 \cdot 2}{5} (- \frac{1}{2} + \cos (- \frac{π}{3})) - \ln (\frac{| 2 |}{| \sec (- \frac{π}{3}) |})}{5}$

Step 3.17.9

Multiply $5$ by $2$ .

$\frac{\frac{10}{5} (- \frac{1}{2} + \cos (- \frac{π}{3})) - \ln (\frac{| 2 |}{| \sec (- \frac{π}{3}) |})}{5}$

Step 3.17.10

Cancel the common factor of $10$ and $5$ .

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Step 3.17.10.1

Factor $5$ out of $10$ .

$\frac{\frac{5 \cdot 2}{5} (- \frac{1}{2} + \cos (- \frac{π}{3})) - \ln (\frac{| 2 |}{| \sec (- \frac{π}{3}) |})}{5}$

Step 3.17.10.2

Cancel the common factors.

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Step 3.17.10.2.1

Factor $5$ out of $5$ .

$\frac{\frac{5 \cdot 2}{5 (1)} (- \frac{1}{2} + \cos (- \frac{π}{3})) - \ln (\frac{| 2 |}{| \sec (- \frac{π}{3}) |})}{5}$

Step 3.17.10.2.2

Cancel the common factor.

$\frac{\frac{5 \cdot 2}{5 \cdot 1} (- \frac{1}{2} + \cos (- \frac{π}{3})) - \ln (\frac{| 2 |}{| \sec (- \frac{π}{3}) |})}{5}$

Step 3.17.10.2.3

Rewrite the expression.

$\frac{\frac{2}{1} (- \frac{1}{2} + \cos (- \frac{π}{3})) - \ln (\frac{| 2 |}{| \sec (- \frac{π}{3}) |})}{5}$

Step 3.17.10.2.4

Divide $2$ by $1$ .

$\frac{2 (- \frac{1}{2} + \cos (- \frac{π}{3})) - \ln (\frac{| 2 |}{| \sec (- \frac{π}{3}) |})}{5}$

Step 3.18

Simplify.

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Step 3.18.1

Simplify each term.

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Step 3.18.1.1

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

$\frac{2 (- \frac{1}{2} + \cos (\frac{5 π}{3})) - \ln (\frac{| 2 |}{| \sec (- \frac{π}{3}) |})}{5}$

Step 3.18.1.2

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

$\frac{2 (- \frac{1}{2} + \cos (\frac{π}{3})) - \ln (\frac{| 2 |}{| \sec (- \frac{π}{3}) |})}{5}$

Step 3.18.1.3

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

$\frac{2 (- \frac{1}{2} + \frac{1}{2}) - \ln (\frac{| 2 |}{| \sec (- \frac{π}{3}) |})}{5}$

Step 3.18.2

Combine the numerators over the common denominator.

$\frac{2 \frac{- 1 + 1}{2} - \ln (\frac{| 2 |}{| \sec (- \frac{π}{3}) |})}{5}$

Step 3.18.3

Add $- 1$ and $1$ .

$\frac{2 (\frac{0}{2}) - \ln (\frac{| 2 |}{| \sec (- \frac{π}{3}) |})}{5}$

Step 3.18.4

Cancel the common factor of $2$ .

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Step 3.18.4.1

Cancel the common factor.

$\frac{2 (\frac{0}{2}) - \ln (\frac{| 2 |}{| \sec (- \frac{π}{3}) |})}{5}$

Step 3.18.4.2

Rewrite the expression.

$\frac{0 - \ln (\frac{| 2 |}{| \sec (- \frac{π}{3}) |})}{5}$

Step 3.18.5

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{0 - \ln (\frac{2}{| \sec (- \frac{π}{3}) |})}{5}$

Step 3.18.6

Simplify the denominator.

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Step 3.18.6.1

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

$\frac{0 - \ln (\frac{2}{| \sec (\frac{5 π}{3}) |})}{5}$

Step 3.18.6.2

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

$\frac{0 - \ln (\frac{2}{| \sec (\frac{π}{3}) |})}{5}$

Step 3.18.6.3

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

$\frac{0 - \ln (\frac{2}{| 2 |})}{5}$

Step 3.18.6.4

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{0 - \ln (\frac{2}{2})}{5}$

Step 3.18.7

Divide $2$ by $2$ .

$\frac{0 - \ln (1)}{5}$

Step 3.18.8

The natural logarithm of $1$ is $0$ .

$\frac{0 - 0}{5}$

Step 3.18.9

Multiply $- 1$ by $0$ .

$\frac{0 + 0}{5}$

Step 3.18.10

Reduce the expression $\frac{0 + 0}{5}$ by cancelling the common factors.

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Step 3.18.10.1

Factor $5$ out of $0$ .

$\frac{5 (0) + 0}{5}$

Step 3.18.10.2

Factor $5$ out of $0$ .

$\frac{5 \cdot 0 + 5 \cdot 0}{5}$

Step 3.18.10.3

Factor $5$ out of $5 \cdot 0 + 5 \cdot 0$ .

$\frac{5 \cdot (0 + 0)}{5}$

Step 3.18.10.4

Factor $5$ out of $5$ .

$\frac{5 \cdot (0 + 0)}{5 (1)}$

Step 3.18.10.5

Cancel the common factor.

$\frac{5 \cdot (0 + 0)}{5 \cdot 1}$

Step 3.18.10.6

Rewrite the expression.

$\frac{0 + 0}{1}$

Step 3.18.11

Divide $0 + 0$ by $1$ .

$0 + 0$

Step 3.18.12

Add $0$ and $0$ .

$0$

Step 4