Calculus Examples

$\int_{0}^{\frac{π}{6}} \cos^{- 5} (2 x) \sin (2 x) d x$

Step 1

Let $u_{2} = \cos (2 x)$ . Then $d u_{2} = - 2 \sin (2 x) d x$ , so $- \frac{1}{2} d u_{2} = \sin (2 x) d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 1.1

Let $u_{2} = \cos (2 x)$ . Find $\frac{d u_{2}}{d x}$ .

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Step 1.1.1

Differentiate $\cos (2 x)$ .

$\frac{d}{d x} [\cos (2 x)]$

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 2 x$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u_{1}$ as $2 x$ .

$\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d x} [2 x]$

Step 1.1.2.2

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

$- \sin (u_{1}) \frac{d}{d x} [2 x]$

Step 1.1.2.3

Replace all occurrences of $u_{1}$ with $2 x$ .

$- \sin (2 x) \frac{d}{d x} [2 x]$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

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

$- \sin (2 x) (2 \frac{d}{d x} [x])$

Step 1.1.3.2

Multiply $2$ by $- 1$ .

$- 2 \sin (2 x) \frac{d}{d x} [x]$

Step 1.1.3.3

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

$- 2 \sin (2 x) \cdot 1$

Step 1.1.3.4

Multiply $- 2$ by $1$ .

$- 2 \sin (2 x)$

Step 1.2

Substitute the lower limit in for $x$ in $u_{2} = \cos (2 x)$ .

$u_{lower} = \cos (2 \cdot 0)$

Step 1.3

Simplify.

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Step 1.3.1

Multiply $2$ by $0$ .

$u_{lower} = \cos (0)$

Step 1.3.2

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

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $x$ in $u_{2} = \cos (2 x)$ .

$u_{upper} = \cos (2 \frac{π}{6})$

Step 1.5

Simplify.

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Step 1.5.1

Cancel the common factor of $2$ .

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Step 1.5.1.1

Factor $2$ out of $6$ .

$u_{upper} = \cos (2 \frac{π}{2 (3)})$

Step 1.5.1.2

Cancel the common factor.

$u_{upper} = \cos (2 \frac{π}{2 \cdot 3})$

Step 1.5.1.3

Rewrite the expression.

$u_{upper} = \cos (\frac{π}{3})$

Step 1.5.2

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

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

Step 1.6

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

$u_{lower} = 1$

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

Step 1.7

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

$\int_{1}^{\frac{1}{2}} {u_{2}}^{- 5} \frac{1}{- 2} d u_{2}$

Step 2

Simplify.

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Step 2.1

Move the negative in front of the fraction.

$\int_{1}^{\frac{1}{2}} {u_{2}}^{- 5} (- \frac{1}{2}) d u_{2}$

Step 2.2

Combine ${u_{2}}^{- 5}$ and $\frac{1}{2}$ .

$\int_{1}^{\frac{1}{2}} - \frac{{u_{2}}^{- 5}}{2} d u_{2}$

Step 2.3

Move ${u_{2}}^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\int_{1}^{\frac{1}{2}} - \frac{1}{2 {u_{2}}^{5}} d u_{2}$

Step 3

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

$- \int_{1}^{\frac{1}{2}} \frac{1}{2 {u_{2}}^{5}} d u_{2}$

Step 4

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

$- (\frac{1}{2} \int_{1}^{\frac{1}{2}} \frac{1}{{u_{2}}^{5}} d u_{2})$

Step 5

Apply basic rules of exponents.

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Step 5.1

Move ${u_{2}}^{5}$ out of the denominator by raising it to the $- 1$ power.

$- \frac{1}{2} \int_{1}^{\frac{1}{2}} {({u_{2}}^{5})}^{- 1} d u_{2}$

Step 5.2

Multiply the exponents in ${({u_{2}}^{5})}^{- 1}$ .

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Step 5.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{1}{2} \int_{1}^{\frac{1}{2}} {u_{2}}^{5 \cdot - 1} d u_{2}$

Step 5.2.2

Multiply $5$ by $- 1$ .

$- \frac{1}{2} \int_{1}^{\frac{1}{2}} {u_{2}}^{- 5} d u_{2}$

Step 6

By the Power Rule, the integral of ${u_{2}}^{- 5}$ with respect to $u_{2}$ is $- \frac{1}{4} {u_{2}}^{- 4}$ .

$- \frac{1}{2} - \frac{1}{4} {u_{2}}^{- 4}]_{1}^{\frac{1}{2}}$

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $- \frac{1}{4} {u_{2}}^{- 4}$ at $\frac{1}{2}$ and at $1$ .

$- \frac{1}{2} ((- \frac{1}{4} {(\frac{1}{2})}^{- 4}) + \frac{1}{4} \cdot 1^{- 4})$

Step 7.2

Simplify.

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Step 7.2.1

Change the sign of the exponent by rewriting the base as its reciprocal.

$- \frac{1}{2} (- \frac{1}{4} \cdot 2^{4} + \frac{1}{4} \cdot 1^{- 4})$

Step 7.2.2

Raise $2$ to the power of $4$ .

$- \frac{1}{2} (- \frac{1}{4} \cdot 16 + \frac{1}{4} \cdot 1^{- 4})$

Step 7.2.3

Multiply $16$ by $- 1$ .

$- \frac{1}{2} (- 16 (\frac{1}{4}) + \frac{1}{4} \cdot 1^{- 4})$

Step 7.2.4

Combine $- 16$ and $\frac{1}{4}$ .

$- \frac{1}{2} (\frac{- 16}{4} + \frac{1}{4} \cdot 1^{- 4})$

Step 7.2.5

Cancel the common factor of $- 16$ and $4$ .

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Step 7.2.5.1

Factor $4$ out of $- 16$ .

$- \frac{1}{2} (\frac{4 \cdot - 4}{4} + \frac{1}{4} \cdot 1^{- 4})$

Step 7.2.5.2

Cancel the common factors.

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Step 7.2.5.2.1

Factor $4$ out of $4$ .

$- \frac{1}{2} (\frac{4 \cdot - 4}{4 (1)} + \frac{1}{4} \cdot 1^{- 4})$

Step 7.2.5.2.2

Cancel the common factor.

$- \frac{1}{2} (\frac{4 \cdot - 4}{4 \cdot 1} + \frac{1}{4} \cdot 1^{- 4})$

Step 7.2.5.2.3

Rewrite the expression.

$- \frac{1}{2} (\frac{- 4}{1} + \frac{1}{4} \cdot 1^{- 4})$

Step 7.2.5.2.4

Divide $- 4$ by $1$ .

$- \frac{1}{2} (- 4 + \frac{1}{4} \cdot 1^{- 4})$

Step 7.2.6

One to any power is one.

$- \frac{1}{2} (- 4 + \frac{1}{4} \cdot 1)$

Step 7.2.7

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

$- \frac{1}{2} (- 4 + \frac{1}{4})$

Step 7.2.8

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

$- \frac{1}{2} (- 4 \cdot \frac{4}{4} + \frac{1}{4})$

Step 7.2.9

Combine $- 4$ and $\frac{4}{4}$ .

$- \frac{1}{2} (\frac{- 4 \cdot 4}{4} + \frac{1}{4})$

Step 7.2.10

Combine the numerators over the common denominator.

$- \frac{1}{2} \cdot \frac{- 4 \cdot 4 + 1}{4}$

Step 7.2.11

Simplify the numerator.

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Step 7.2.11.1

Multiply $- 4$ by $4$ .

$- \frac{1}{2} \cdot \frac{- 16 + 1}{4}$

Step 7.2.11.2

Add $- 16$ and $1$ .

$- \frac{1}{2} \cdot \frac{- 15}{4}$

Step 7.2.12

Move the negative in front of the fraction.

$- \frac{1}{2} (- \frac{15}{4})$

Step 7.2.13

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{2}) \frac{15}{4}$

Step 7.2.14

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

$\frac{1}{2} \cdot \frac{15}{4}$

Step 7.2.15

Multiply $\frac{1}{2}$ by $\frac{15}{4}$ .

$\frac{15}{2 \cdot 4}$

Step 7.2.16

Multiply $2$ by $4$ .

$\frac{15}{8}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{15}{8}$

Decimal Form:

$1.875$

Mixed Number Form:

$1 \frac{7}{8}$