Calculus Examples

$\int_{0}^{1} \sqrt{x - x^{2}} d x$

Step 1

Complete the square.

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Step 1.1

Reorder $x$ and $- x^{2}$ .

$- x^{2} + x$

Step 1.2

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - 1$

$b = 1$

$c = 0$

Step 1.3

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.4

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.4.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{1}{2 \cdot - 1}$

Step 1.4.2

Cancel the common factor of $1$ and $- 1$ .

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Step 1.4.2.1

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

$d = \frac{- 1 (- 1)}{2 \cdot - 1}$

Step 1.4.2.2

Move the negative in front of the fraction.

$d = - \frac{1}{2}$

Step 1.5

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.5.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{1^{2}}{4 \cdot - 1}$

Step 1.5.2

Simplify the right side.

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Step 1.5.2.1

Simplify each term.

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Step 1.5.2.1.1

One to any power is one.

$e = 0 - \frac{1}{4 \cdot - 1}$

Step 1.5.2.1.2

Multiply $4$ by $- 1$ .

$e = 0 - \frac{1}{- 4}$

Step 1.5.2.1.3

Move the negative in front of the fraction.

$e = 0 - - \frac{1}{4}$

Step 1.5.2.1.4

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

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Step 1.5.2.1.4.1

Multiply $- 1$ by $- 1$ .

$e = 0 + 1 (\frac{1}{4})$

Step 1.5.2.1.4.2

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

$e = 0 + \frac{1}{4}$

Step 1.5.2.2

Add $0$ and $\frac{1}{4}$ .

$e = \frac{1}{4}$

Step 1.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x - \frac{1}{2})}^{2} + \frac{1}{4}$ .

$\int_{0}^{1} \sqrt{- {(x - \frac{1}{2})}^{2} + \frac{1}{4}} d x$

Step 2

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

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Step 2.1

Let $u_{1} = x - \frac{1}{2}$ . Find $\frac{d u_{1}}{d x}$ .

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Step 2.1.1

Differentiate $x - \frac{1}{2}$ .

$\frac{d}{d x} [x - \frac{1}{2}]$

Step 2.1.2

By the Sum Rule, the derivative of $x - \frac{1}{2}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{1}{2}]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{1}{2}]$

Step 2.1.3

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

$1 + \frac{d}{d x} [- \frac{1}{2}]$

Step 2.1.4

Since $- \frac{1}{2}$ is constant with respect to $x$ , the derivative of $- \frac{1}{2}$ with respect to $x$ is $0$ .

$1 + 0$

Step 2.1.5

Add $1$ and $0$ .

$1$

Step 2.2

Substitute the lower limit in for $x$ in $u_{1} = x - \frac{1}{2}$ .

$u_{lower} = 0 - \frac{1}{2}$

Step 2.3

Subtract $\frac{1}{2}$ from $0$ .

$u_{lower} = - \frac{1}{2}$

Step 2.4

Substitute the upper limit in for $x$ in $u_{1} = x - \frac{1}{2}$ .

$u_{upper} = 1 - \frac{1}{2}$

Step 2.5

Simplify.

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Step 2.5.1

Write $1$ as a fraction with a common denominator.

$u_{upper} = \frac{2}{2} - \frac{1}{2}$

Step 2.5.2

Combine the numerators over the common denominator.

$u_{upper} = \frac{2 - 1}{2}$

Step 2.5.3

Subtract $1$ from $2$ .

$u_{upper} = \frac{1}{2}$

Step 2.6

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

$u_{lower} = - \frac{1}{2}$

$u_{upper} = \frac{1}{2}$

Step 2.7

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

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \sqrt{- {u_{1}}^{2} + \frac{1}{4}} d u_{1}$

Step 3

Write the expression using exponents.

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Step 3.1

Rewrite $1$ as $1^{2}$ .

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \sqrt{- {u_{1}}^{2} + \frac{1^{2}}{4}} d u_{1}$

Step 3.2

Rewrite $4$ as $2^{2}$ .

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \sqrt{- {u_{1}}^{2} + \frac{1^{2}}{2^{2}}} d u_{1}$

Step 4

Rewrite $\frac{1^{2}}{2^{2}}$ as ${(\frac{1}{2})}^{2}$ .

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \sqrt{- {u_{1}}^{2} + {(\frac{1}{2})}^{2}} d u_{1}$

Step 5

Reorder $- {u_{1}}^{2}$ and ${(\frac{1}{2})}^{2}$ .

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \sqrt{{(\frac{1}{2})}^{2} - {u_{1}}^{2}} d u_{1}$

Step 6

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{1}{2}$ and $b = u_{1}$ .

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \sqrt{(\frac{1}{2} + u_{1}) (\frac{1}{2} - u_{1})} d u_{1}$

Step 7

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

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \sqrt{(\frac{1}{2} + u_{1} \cdot \frac{2}{2}) (\frac{1}{2} - u_{1})} d u_{1}$

Step 8

Simplify terms.

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Step 8.1

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

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \sqrt{(\frac{1}{2} + \frac{u_{1} \cdot 2}{2}) (\frac{1}{2} - u_{1})} d u_{1}$

Step 8.2

Combine the numerators over the common denominator.

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \sqrt{\frac{1 + u_{1} \cdot 2}{2} (\frac{1}{2} - u_{1})} d u_{1}$

Step 9

Move $2$ to the left of $u_{1}$ .

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \sqrt{\frac{1 + 2 u_{1}}{2} (\frac{1}{2} - u_{1})} d u_{1}$

Step 10

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

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \sqrt{\frac{1 + 2 u_{1}}{2} (\frac{1}{2} - u_{1} \cdot \frac{2}{2})} d u_{1}$

Step 11

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

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \sqrt{\frac{1 + 2 u_{1}}{2} (\frac{1}{2} + \frac{- u_{1} \cdot 2}{2})} d u_{1}$

Step 12

Combine the numerators over the common denominator.

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \sqrt{\frac{1 + 2 u_{1}}{2} \cdot \frac{1 - u_{1} \cdot 2}{2}} d u_{1}$

Step 13

Multiply $2$ by $- 1$ .

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \sqrt{\frac{1 + 2 u_{1}}{2} \cdot \frac{1 - 2 u_{1}}{2}} d u_{1}$

Step 14

Multiply $\frac{1 + 2 u_{1}}{2}$ by $\frac{1 - 2 u_{1}}{2}$ .

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \sqrt{\frac{(1 + 2 u_{1}) (1 - 2 u_{1})}{2 \cdot 2}} d u_{1}$

Step 15

Multiply $2$ by $2$ .

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \sqrt{\frac{(1 + 2 u_{1}) (1 - 2 u_{1})}{4}} d u_{1}$

Step 16

Rewrite $\frac{(1 + 2 u_{1}) (1 - 2 u_{1})}{4}$ as ${(\frac{1}{2})}^{2} ((1 + 2 u_{1}) (1 - 2 u_{1}))$ .

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Step 16.1

Factor the perfect power $1^{2}$ out of $(1 + 2 u_{1}) (1 - 2 u_{1})$ .

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \sqrt{\frac{1^{2} ((1 + 2 u_{1}) (1 - 2 u_{1}))}{4}} d u_{1}$

Step 16.2

Factor the perfect power $2^{2}$ out of $4$ .

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \sqrt{\frac{1^{2} ((1 + 2 u_{1}) (1 - 2 u_{1}))}{2^{2} \cdot 1}} d u_{1}$

Step 16.3

Rearrange the fraction $\frac{1^{2} ((1 + 2 u_{1}) (1 - 2 u_{1}))}{2^{2} \cdot 1}$ .

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \sqrt{{(\frac{1}{2})}^{2} ((1 + 2 u_{1}) (1 - 2 u_{1}))} d u_{1}$

Step 17

Simplify terms.

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Step 17.1

Pull terms out from under the radical.

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{1}{2} \sqrt{(1 + 2 u_{1}) (1 - 2 u_{1})} d u_{1}$

Step 17.2

Combine $\frac{1}{2}$ and $\sqrt{(1 + 2 u_{1}) (1 - 2 u_{1})}$ .

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{\sqrt{(1 + 2 u_{1}) (1 - 2 u_{1})}}{2} d u_{1}$

Step 18

Expand $(1 + 2 u_{1}) (1 - 2 u_{1})$ using the FOIL Method.

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Step 18.1

Apply the distributive property.

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{\sqrt{1 (1 - 2 u_{1}) + 2 u_{1} (1 - 2 u_{1})}}{2} d u_{1}$

Step 18.2

Apply the distributive property.

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{\sqrt{1 \cdot 1 + 1 (- 2 u_{1}) + 2 u_{1} (1 - 2 u_{1})}}{2} d u_{1}$

Step 18.3

Apply the distributive property.

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{\sqrt{1 \cdot 1 + 1 (- 2 u_{1}) + 2 u_{1} \cdot 1 + 2 u_{1} (- 2 u_{1})}}{2} d u_{1}$

Step 19

Simplify and combine like terms.

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Step 19.1

Simplify each term.

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Step 19.1.1

Multiply $1$ by $1$ .

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{\sqrt{1 + 1 (- 2 u_{1}) + 2 u_{1} \cdot 1 + 2 u_{1} (- 2 u_{1})}}{2} d u_{1}$

Step 19.1.2

Multiply $- 2 u_{1}$ by $1$ .

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{\sqrt{1 - 2 u_{1} + 2 u_{1} \cdot 1 + 2 u_{1} (- 2 u_{1})}}{2} d u_{1}$

Step 19.1.3

Multiply $2$ by $1$ .

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{\sqrt{1 - 2 u_{1} + 2 u_{1} + 2 u_{1} (- 2 u_{1})}}{2} d u_{1}$

Step 19.1.4

Rewrite using the commutative property of multiplication.

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{\sqrt{1 - 2 u_{1} + 2 u_{1} + 2 \cdot - 2 u_{1} \cdot u_{1}}}{2} d u_{1}$

Step 19.1.5

Multiply $u_{1}$ by $u_{1}$ by adding the exponents.

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Step 19.1.5.1

Move $u_{1}$ .

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{\sqrt{1 - 2 u_{1} + 2 u_{1} + 2 \cdot - 2 (u_{1} \cdot u_{1})}}{2} d u_{1}$

Step 19.1.5.2

Multiply $u_{1}$ by $u_{1}$ .

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{\sqrt{1 - 2 u_{1} + 2 u_{1} + 2 \cdot - 2 {u_{1}}^{2}}}{2} d u_{1}$

Step 19.1.6

Multiply $2$ by $- 2$ .

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{\sqrt{1 - 2 u_{1} + 2 u_{1} - 4 {u_{1}}^{2}}}{2} d u_{1}$

Step 19.2

Add $- 2 u_{1}$ and $2 u_{1}$ .

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{\sqrt{1 + 0 - 4 {u_{1}}^{2}}}{2} d u_{1}$

Step 19.3

Add $1$ and $0$ .

$\int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{\sqrt{1 - 4 {u_{1}}^{2}}}{2} d u_{1}$

Step 20

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

$\frac{1}{2} \int_{- \frac{1}{2}}^{\frac{1}{2}} \sqrt{1 - 4 {u_{1}}^{2}} d u_{1}$

Step 21

Let $u_{1} = \frac{1}{2} \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d u_{1} = \frac{\cos (t)}{2} d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\frac{\cos (t)}{2}$ is positive.

$\frac{1}{2} \int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{1 - 4 {(\frac{1}{2} \sin (t))}^{2}} \frac{\cos (t)}{2} d t$

Step 22

Simplify terms.

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Step 22.1

Simplify $\sqrt{1 - 4 {(\frac{1}{2} \sin (t))}^{2}}$ .

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Step 22.1.1

Simplify each term.

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Step 22.1.1.1

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

$\frac{1}{2} \int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{1 - 4 {(\frac{\sin (t)}{2})}^{2}} \frac{\cos (t)}{2} d t$

Step 22.1.1.2

Apply the product rule to $\frac{\sin (t)}{2}$ .

$\frac{1}{2} \int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{1 - 4 \frac{\sin^{2} (t)}{2^{2}}} \frac{\cos (t)}{2} d t$

Step 22.1.1.3

Raise $2$ to the power of $2$ .

$\frac{1}{2} \int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{1 - 4 \frac{\sin^{2} (t)}{4}} \frac{\cos (t)}{2} d t$

Step 22.1.1.4

Cancel the common factor of $4$ .

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Step 22.1.1.4.1

Factor $4$ out of $- 4$ .

$\frac{1}{2} \int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{1 + 4 (- 1) \frac{\sin^{2} (t)}{4}} \frac{\cos (t)}{2} d t$

Step 22.1.1.4.2

Cancel the common factor.

$\frac{1}{2} \int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{1 + 4 \cdot - 1 \frac{\sin^{2} (t)}{4}} \frac{\cos (t)}{2} d t$

Step 22.1.1.4.3

Rewrite the expression.

$\frac{1}{2} \int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{1 - \sin^{2} (t)} \frac{\cos (t)}{2} d t$

Step 22.1.2

Apply pythagorean identity.

$\frac{1}{2} \int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{\cos^{2} (t)} \frac{\cos (t)}{2} d t$

Step 22.1.3

Pull terms out from under the radical, assuming positive real numbers.

$\frac{1}{2} \int_{- \frac{π}{2}}^{\frac{π}{2}} \cos (t) \frac{\cos (t)}{2} d t$

Step 22.2

Simplify.

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Step 22.2.1

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

$\frac{1}{2} \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{\cos (t) \cos (t)}{2} d t$

Step 22.2.2

Raise $\cos (t)$ to the power of $1$ .

$\frac{1}{2} \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{\cos^{1} (t) \cos (t)}{2} d t$

Step 22.2.3

Raise $\cos (t)$ to the power of $1$ .

$\frac{1}{2} \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{\cos^{1} (t) \cos^{1} (t)}{2} d t$

Step 22.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{2} \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{{\cos (t)}^{1 + 1}}{2} d t$

Step 22.2.5

Add $1$ and $1$ .

$\frac{1}{2} \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{\cos^{2} (t)}{2} d t$

Step 23

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

$\frac{1}{2} (\frac{1}{2} \int_{- \frac{π}{2}}^{\frac{π}{2}} \cos^{2} (t) d t)$

Step 24

Simplify.

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Step 24.1

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

$\frac{1}{2 \cdot 2} \int_{- \frac{π}{2}}^{\frac{π}{2}} \cos^{2} (t) d t$

Step 24.2

Multiply $2$ by $2$ .

$\frac{1}{4} \int_{- \frac{π}{2}}^{\frac{π}{2}} \cos^{2} (t) d t$

Step 25

Use the half-angle formula to rewrite $\cos^{2} (t)$ as $\frac{1 + \cos (2 t)}{2}$ .

$\frac{1}{4} \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{1 + \cos (2 t)}{2} d t$

Step 26

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

$\frac{1}{4} (\frac{1}{2} \int_{- \frac{π}{2}}^{\frac{π}{2}} 1 + \cos (2 t) d t)$

Step 27

Simplify.

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Step 27.1

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

$\frac{1}{2 \cdot 4} \int_{- \frac{π}{2}}^{\frac{π}{2}} 1 + \cos (2 t) d t$

Step 27.2

Multiply $2$ by $4$ .

$\frac{1}{8} \int_{- \frac{π}{2}}^{\frac{π}{2}} 1 + \cos (2 t) d t$

Step 28

Split the single integral into multiple integrals.

$\frac{1}{8} (\int_{- \frac{π}{2}}^{\frac{π}{2}} d t + \int_{- \frac{π}{2}}^{\frac{π}{2}} \cos (2 t) d t)$

Step 29

Apply the constant rule.

$\frac{1}{8} (t]_{- \frac{π}{2}}^{\frac{π}{2}} + \int_{- \frac{π}{2}}^{\frac{π}{2}} \cos (2 t) d t)$

Step 30

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

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Step 30.1

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

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Step 30.1.1

Differentiate $2 t$ .

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

Step 30.1.2

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

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

Step 30.1.3

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

$2 \cdot 1$

Step 30.1.4

Multiply $2$ by $1$ .

$2$

Step 30.2

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

$u_{lower} = 2 (- \frac{π}{2})$

Step 30.3

Cancel the common factor of $2$ .

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Step 30.3.1

Move the leading negative in $- \frac{π}{2}$ into the numerator.

$u_{lower} = 2 \frac{- π}{2}$

Step 30.3.2

Cancel the common factor.

$u_{lower} = 2 \frac{- π}{2}$

Step 30.3.3

Rewrite the expression.

$u_{lower} = - π$

Step 30.4

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

$u_{upper} = 2 \frac{π}{2}$

Step 30.5

Cancel the common factor of $2$ .

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Step 30.5.1

Cancel the common factor.

$u_{upper} = 2 \frac{π}{2}$

Step 30.5.2

Rewrite the expression.

$u_{upper} = π$

Step 30.6

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

$u_{lower} = - π$

$u_{upper} = π$

Step 30.7

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

$\frac{1}{8} (t]_{- \frac{π}{2}}^{\frac{π}{2}} + \int_{- π}^{π} \cos (u_{2}) \frac{1}{2} d u_{2})$

Step 31

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

$\frac{1}{8} (t]_{- \frac{π}{2}}^{\frac{π}{2}} + \int_{- π}^{π} \frac{\cos (u_{2})}{2} d u_{2})$

Step 32

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

$\frac{1}{8} (t]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{1}{2} \int_{- π}^{π} \cos (u_{2}) d u_{2})$

Step 33

The integral of $\cos (u_{2})$ with respect to $u_{2}$ is $\sin (u_{2})$ .

$\frac{1}{8} (t]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{1}{2} \sin (u_{2})]_{- π}^{π})$

Step 34

Substitute and simplify.

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Step 34.1

Evaluate $t$ at $\frac{π}{2}$ and at $- \frac{π}{2}$ .

$\frac{1}{8} ((\frac{π}{2}) + \frac{π}{2} + \frac{1}{2} \sin (u_{2})]_{- π}^{π})$

Step 34.2

Evaluate $\sin (u_{2})$ at $π$ and at $- π$ .

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

Step 34.3

Simplify.

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Step 34.3.1

Combine the numerators over the common denominator.

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

Step 34.3.2

Add $π$ and $π$ .

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

Step 34.3.3

Cancel the common factor of $2$ .

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Step 34.3.3.1

Cancel the common factor.

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

Step 34.3.3.2

Divide $π$ by $1$ .

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

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

Step 35

Simplify.

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Step 35.1

Simplify each term.

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Step 35.1.1

Simplify each term.

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Step 35.1.1.1

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

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

Step 35.1.1.2

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

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

Step 35.1.1.3

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

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

Step 35.1.1.4

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

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

Step 35.1.1.5

Multiply $- 1$ by $0$ .

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

Step 35.1.2

Add $0$ and $0$ .

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

Step 35.1.3

Multiply $\frac{1}{2}$ by $0$ .

$\frac{1}{8} (π + 0)$

Step 35.2

Add $π$ and $0$ .

$\frac{1}{8} π$

Step 35.3

Combine $\frac{1}{8}$ and $π$ .

$\frac{π}{8}$

Step 36

The result can be shown in multiple forms.

Exact Form:

$\frac{π}{8}$

Decimal Form:

$0.39269908 \dots$

Step 37