Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $\sin (2 x)$ .

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

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 $f (x) = \sin (x)$ and $g (x) = 2 x$ .

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

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 $2 \frac{d}{d x} [x]$ .

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

Step 1.1.3.2

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

$\cos (2 x) (2 \cdot 1)$

Step 1.1.3.3

Simplify the expression.

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Step 1.1.3.3.1

Multiply $2$ by $1$ .

$\cos (2 x) \cdot 2$

Step 1.1.3.3.2

Move $2$ to the left of $\cos (2 x)$ .

$2 \cos (2 x)$

Step 1.2

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

$u_{lower} = \sin (2 \frac{π}{6})$

Step 1.3

Simplify.

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Step 1.3.1

Cancel the common factor of $2$ .

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Step 1.3.1.1

Factor $2$ out of $6$ .

$u_{lower} = \sin (2 \frac{π}{2 (3)})$

Step 1.3.1.2

Cancel the common factor.

$u_{lower} = \sin (2 \frac{π}{2 \cdot 3})$

Step 1.3.1.3

Rewrite the expression.

$u_{lower} = \sin (\frac{π}{3})$

Step 1.3.2

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

$u_{lower} = \frac{\sqrt{3}}{2}$

Step 1.4

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

$u_{upper} = \sin (2 \frac{π}{2})$

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

Cancel the common factor.

$u_{upper} = \sin (2 \frac{π}{2})$

Step 1.5.1.2

Rewrite the expression.

$u_{upper} = \sin (π)$

Step 1.5.2

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

$u_{upper} = \sin (0)$

Step 1.5.3

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

$u_{upper} = 0$

Step 1.6

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

$u_{lower} = \frac{\sqrt{3}}{2}$

$u_{upper} = 0$

Step 1.7

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

$\int_{\frac{\sqrt{3}}{2}}^{0} {u_{2}}^{5} \frac{1}{2} d u_{2}$

Step 2

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

$\int_{\frac{\sqrt{3}}{2}}^{0} \frac{{u_{2}}^{5}}{2} d u_{2}$

Step 3

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

$\frac{1}{2} \int_{\frac{\sqrt{3}}{2}}^{0} {u_{2}}^{5} d u_{2}$

Step 4

By the Power Rule, the integral of ${u_{2}}^{5}$ with respect to $u_{2}$ is $\frac{1}{6} {u_{2}}^{6}$ .

$\frac{1}{2} \frac{1}{6} {u_{2}}^{6}]_{\frac{\sqrt{3}}{2}}^{0}$

Step 5

Simplify the expression.

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Step 5.1

Evaluate $\frac{1}{6} {u_{2}}^{6}$ at $0$ and at $\frac{\sqrt{3}}{2}$ .

$\frac{1}{2} ((\frac{1}{6} \cdot 0^{6}) - \frac{1}{6} {(\frac{\sqrt{3}}{2})}^{6})$

Step 5.2

Simplify the expression.

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Step 5.2.1

Raising $0$ to any positive power yields $0$ .

$\frac{1}{2} (\frac{1}{6} \cdot 0 - \frac{1}{6} {(\frac{\sqrt{3}}{2})}^{6})$

Step 5.2.2

Multiply $\frac{1}{6}$ by $0$ .

$\frac{1}{2} (0 - \frac{1}{6} {(\frac{\sqrt{3}}{2})}^{6})$

Step 5.2.3

Subtract $\frac{1}{6} {(\frac{\sqrt{3}}{2})}^{6}$ from $0$ .

$\frac{1}{2} (- \frac{1}{6} {(\frac{\sqrt{3}}{2})}^{6})$

Step 5.3

Simplify.

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Step 5.3.1

Multiply $\frac{1}{2}$ by $\frac{1}{6}$ .

$- \frac{1}{2 \cdot 6} {(\frac{\sqrt{3}}{2})}^{6}$

Step 5.3.2

Multiply $2$ by $6$ .

$- \frac{1}{12} {(\frac{\sqrt{3}}{2})}^{6}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{12} \cdot {(\frac{\sqrt{3}}{2})}^{6}$

Decimal Form:

$- 0.03515625$