Calculus Examples

$\int_{0}^{π} 5 {(5 - 4 \cos (t))}^{\frac{1}{4}} \sin (t) d t$

Step 1

Since $5$ is constant with respect to $t$ , move $5$ out of the integral.

$5 \int_{0}^{π} {(5 - 4 \cos (t))}^{\frac{1}{4}} \sin (t) d t$

Step 2

Let $u = 5 - 4 \cos (t)$ . Then $d u = 4 \sin (t) d t$ , so $\frac{1}{4} d u = \sin (t) d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 5 - 4 \cos (t)$ . Find $\frac{d u}{d t}$ .

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Step 2.1.1

Differentiate $5 - 4 \cos (t)$ .

$\frac{d}{d t} [5 - 4 \cos (t)]$

Step 2.1.2

Differentiate.

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Step 2.1.2.1

By the Sum Rule, the derivative of $5 - 4 \cos (t)$ with respect to $t$ is $\frac{d}{d t} [5] + \frac{d}{d t} [- 4 \cos (t)]$ .

$\frac{d}{d t} [5] + \frac{d}{d t} [- 4 \cos (t)]$

Step 2.1.2.2

Since $5$ is constant with respect to $t$ , the derivative of $5$ with respect to $t$ is $0$ .

$0 + \frac{d}{d t} [- 4 \cos (t)]$

Step 2.1.3

Evaluate $\frac{d}{d t} [- 4 \cos (t)]$ .

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Step 2.1.3.1

Since $- 4$ is constant with respect to $t$ , the derivative of $- 4 \cos (t)$ with respect to $t$ is $- 4 \frac{d}{d t} [\cos (t)]$ .

$0 - 4 \frac{d}{d t} [\cos (t)]$

Step 2.1.3.2

The derivative of $\cos (t)$ with respect to $t$ is $- \sin (t)$ .

$0 - 4 (- \sin (t))$

Step 2.1.3.3

Multiply $- 1$ by $- 4$ .

$0 + 4 \sin (t)$

Step 2.1.4

Add $0$ and $4 \sin (t)$ .

$4 \sin (t)$

Step 2.2

Substitute the lower limit in for $t$ in $u = 5 - 4 \cos (t)$ .

$u_{lower} = 5 - 4 \cos (0)$

Step 2.3

Simplify.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

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

$u_{lower} = 5 - 4 \cdot 1$

Step 2.3.1.2

Multiply $- 4$ by $1$ .

$u_{lower} = 5 - 4$

Step 2.3.2

Subtract $4$ from $5$ .

$u_{lower} = 1$

Step 2.4

Substitute the upper limit in for $t$ in $u = 5 - 4 \cos (t)$ .

$u_{upper} = 5 - 4 \cos (π)$

Step 2.5

Simplify.

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Step 2.5.1

Simplify each term.

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Step 2.5.1.1

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 second quadrant.

$u_{upper} = 5 - 4 (- \cos (0))$

Step 2.5.1.2

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

$u_{upper} = 5 - 4 (- 1 \cdot 1)$

Step 2.5.1.3

Multiply $- 4 (- 1 \cdot 1)$ .

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Step 2.5.1.3.1

Multiply $- 1$ by $1$ .

$u_{upper} = 5 - 4 \cdot - 1$

Step 2.5.1.3.2

Multiply $- 4$ by $- 1$ .

$u_{upper} = 5 + 4$

Step 2.5.2

Add $5$ and $4$ .

$u_{upper} = 9$

Step 2.6

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

$u_{lower} = 1$

$u_{upper} = 9$

Step 2.7

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

$5 \int_{1}^{9} u^{\frac{1}{4}} \frac{1}{4} d u$

Step 3

Combine $u^{\frac{1}{4}}$ and $\frac{1}{4}$ .

$5 \int_{1}^{9} \frac{u^{\frac{1}{4}}}{4} d u$

Step 4

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

$5 (\frac{1}{4} \int_{1}^{9} u^{\frac{1}{4}} d u)$

Step 5

Combine $\frac{1}{4}$ and $5$ .

$\frac{5}{4} \int_{1}^{9} u^{\frac{1}{4}} d u$

Step 6

By the Power Rule, the integral of $u^{\frac{1}{4}}$ with respect to $u$ is $\frac{4}{5} u^{\frac{5}{4}}$ .

$\frac{5}{4} \frac{4}{5} u^{\frac{5}{4}}]_{1}^{9}$

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $\frac{4}{5} u^{\frac{5}{4}}$ at $9$ and at $1$ .

$\frac{5}{4} ((\frac{4}{5} \cdot 9^{\frac{5}{4}}) - \frac{4}{5} \cdot 1^{\frac{5}{4}})$

Step 7.2

Simplify.

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Step 7.2.1

Combine $\frac{4}{5}$ and $9^{\frac{5}{4}}$ .

$\frac{5}{4} (\frac{4 \cdot 9^{\frac{5}{4}}}{5} - \frac{4}{5} \cdot 1^{\frac{5}{4}})$

Step 7.2.2

One to any power is one.

$\frac{5}{4} (\frac{4 \cdot 9^{\frac{5}{4}}}{5} - \frac{4}{5} \cdot 1)$

Step 7.2.3

Multiply $- 1$ by $1$ .

$\frac{5}{4} (\frac{4 \cdot 9^{\frac{5}{4}}}{5} - \frac{4}{5})$

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

$\frac{5}{4} \cdot \frac{4 \cdot 9^{\frac{5}{4}}}{5} + \frac{5}{4} (- \frac{4}{5})$

Step 8.2

Combine.

$\frac{5 (4 \cdot 9^{\frac{5}{4}})}{4 \cdot 5} + \frac{5}{4} (- \frac{4}{5})$

Step 8.3

Cancel the common factor of $5$ .

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Step 8.3.1

Move the leading negative in $- \frac{4}{5}$ into the numerator.

$\frac{5 (4 \cdot 9^{\frac{5}{4}})}{4 \cdot 5} + \frac{5}{4} \cdot \frac{- 4}{5}$

Step 8.3.2

Cancel the common factor.

$\frac{5 (4 \cdot 9^{\frac{5}{4}})}{4 \cdot 5} + \frac{5}{4} \cdot \frac{- 4}{5}$

Step 8.3.3

Rewrite the expression.

$\frac{5 (4 \cdot 9^{\frac{5}{4}})}{4 \cdot 5} + \frac{1}{4} \cdot - 4$

Step 8.4

Cancel the common factor of $4$ .

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Step 8.4.1

Factor $4$ out of $- 4$ .

$\frac{5 (4 \cdot 9^{\frac{5}{4}})}{4 \cdot 5} + \frac{1}{4} \cdot (4 (- 1))$

Step 8.4.2

Cancel the common factor.

$\frac{5 (4 \cdot 9^{\frac{5}{4}})}{4 \cdot 5} + \frac{1}{4} \cdot (4 \cdot - 1)$

Step 8.4.3

Rewrite the expression.

$\frac{5 (4 \cdot 9^{\frac{5}{4}})}{4 \cdot 5} - 1$

Step 8.5

Simplify each term.

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Step 8.5.1

Cancel the common factor.

$\frac{5 (4 \cdot 9^{\frac{5}{4}})}{4 \cdot 5} - 1$

Step 8.5.2

Rewrite the expression.

$\frac{4 \cdot 9^{\frac{5}{4}}}{4} - 1$

Step 8.5.3

Cancel the common factor.

$\frac{4 \cdot 9^{\frac{5}{4}}}{4} - 1$

Step 8.5.4

Divide $9^{\frac{5}{4}}$ by $1$ .

$9^{\frac{5}{4}} - 1$

Step 9

The result can be shown in multiple forms.

Exact Form:

$9^{\frac{5}{4}} - 1$

Decimal Form:

$14.58845726 \dots$