Calculus Examples

$\int_{0}^{\frac{π}{12}} \cos (2 x) \sin (2 x) d x$

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 $u_{2} = \cos (2 x)$ .

$u_{upper} = \cos (2 \frac{π}{12})$

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 $12$ .

$u_{upper} = \cos (2 \frac{π}{2 (6)})$

Step 1.5.1.2

Cancel the common factor.

$u_{upper} = \cos (2 \frac{π}{2 \cdot 6})$

Step 1.5.1.3

Rewrite the expression.

$u_{upper} = \cos (\frac{π}{6})$

Step 1.5.2

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

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

Step 1.6

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

$u_{lower} = 1$

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

Step 1.7

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

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

Step 2

Simplify.

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Step 2.1

Move the negative in front of the fraction.

$\int_{1}^{\frac{\sqrt{3}}{2}} u_{2} (- \frac{1}{2}) d u_{2}$

Step 2.2

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

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

Step 3

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

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

Step 4

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

$- (\frac{1}{2} \int_{1}^{\frac{\sqrt{3}}{2}} u_{2} d u_{2})$

Step 5

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

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

Step 6

Substitute and simplify.

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Step 6.1

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

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

Step 6.2

Simplify.

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Step 6.2.1

One to any power is one.

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

Step 6.2.2

Multiply $- 1$ by $1$ .

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

Step 7

Simplify.

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Step 7.1

Simplify each term.

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Step 7.1.1

Apply the product rule to $\frac{\sqrt{3}}{2}$ .

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

Step 7.1.2

Combine.

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

Step 7.1.3

Multiply $2$ by $2^{2}$ by adding the exponents.

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Step 7.1.3.1

Multiply $2$ by $2^{2}$ .

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Step 7.1.3.1.1

Raise $2$ to the power of $1$ .

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

Step 7.1.3.1.2

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

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

Step 7.1.3.2

Add $1$ and $2$ .

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

Step 7.1.4

Multiply ${\sqrt{3}}^{2}$ by $1$ .

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

Step 7.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 7.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 7.1.5.2

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

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

Step 7.1.5.3

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

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

Step 7.1.5.4

Cancel the common factor of $2$ .

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Step 7.1.5.4.1

Cancel the common factor.

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

Step 7.1.5.4.2

Rewrite the expression.

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

Step 7.1.5.5

Evaluate the exponent.

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

Step 7.1.6

Raise $2$ to the power of $3$ .

$- \frac{1}{2} (\frac{3}{8} - \frac{1}{2})$

Step 7.2

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{1}{2} (\frac{3}{8} - \frac{1}{2} \cdot \frac{4}{4})$

Step 7.3

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.3.1

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

$- \frac{1}{2} (\frac{3}{8} - \frac{4}{2 \cdot 4})$

Step 7.3.2

Multiply $2$ by $4$ .

$- \frac{1}{2} (\frac{3}{8} - \frac{4}{8})$

Step 7.4

Combine the numerators over the common denominator.

$- \frac{1}{2} \cdot \frac{3 - 4}{8}$

Step 7.5

Subtract $4$ from $3$ .

$- \frac{1}{2} \cdot \frac{- 1}{8}$

Step 7.6

Move the negative in front of the fraction.

$- \frac{1}{2} (- \frac{1}{8})$

Step 7.7

Multiply $- \frac{1}{2} (- \frac{1}{8})$ .

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Step 7.7.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{2}) \frac{1}{8}$

Step 7.7.2

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

$\frac{1}{2} \cdot \frac{1}{8}$

Step 7.7.3

Multiply $\frac{1}{2}$ by $\frac{1}{8}$ .

$\frac{1}{2 \cdot 8}$

Step 7.7.4

Multiply $2$ by $8$ .

$\frac{1}{16}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{16}$

Decimal Form:

$0.0625$