Calculus Examples

$\int_{\frac{π}{6}}^{\frac{π}{4}} 2 \sin (2 x) \cos (2 x) d x$

Step 1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int_{\frac{π}{6}}^{\frac{π}{4}} \sin (2 x) \cos (2 x) d x$

Step 2

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 2.1

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

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Step 2.1.1

Differentiate $\sin (2 x)$ .

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

Step 2.1.2

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 2.1.2.1

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 2.1.2.2

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 2.1.2.3

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

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

Step 2.1.3

Differentiate.

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Step 2.1.3.1

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 2.1.3.2

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 2.1.3.3

Simplify the expression.

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Step 2.1.3.3.1

Multiply $2$ by $1$ .

$\cos (2 x) \cdot 2$

Step 2.1.3.3.2

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

$2 \cos (2 x)$

Step 2.2

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

$u_{lower} = \sin (2 \frac{π}{6})$

Step 2.3

Simplify.

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Step 2.3.1

Cancel the common factor of $2$ .

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Step 2.3.1.1

Factor $2$ out of $6$ .

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

Step 2.3.1.2

Cancel the common factor.

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

Step 2.3.1.3

Rewrite the expression.

$u_{lower} = \sin (\frac{π}{3})$

Step 2.3.2

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

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

Step 2.4

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

$u_{upper} = \sin (2 \frac{π}{4})$

Step 2.5

Simplify.

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Step 2.5.1

Cancel the common factor of $2$ .

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Step 2.5.1.1

Factor $2$ out of $4$ .

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

Step 2.5.1.2

Cancel the common factor.

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

Step 2.5.1.3

Rewrite the expression.

$u_{upper} = \sin (\frac{π}{2})$

Step 2.5.2

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

$u_{upper} = 1$

Step 2.6

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

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

$u_{upper} = 1$

Step 2.7

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

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

Cancel the common factor of $2$ .

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Step 5.2.1

Cancel the common factor.

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

Step 5.2.2

Rewrite the expression.

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

Step 5.3

Multiply $\int_{\frac{\sqrt{3}}{2}}^{1} u_{2} d u_{2}$ by $1$ .

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

Step 6

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

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

Step 7

Substitute and simplify.

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Step 7.1

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

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

Step 7.2

Simplify.

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Step 7.2.1

One to any power is one.

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

Step 7.2.2

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

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

Step 8

Simplify.

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Step 8.1

Simplify each term.

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Step 8.1.1

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

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

Step 8.1.2

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

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Step 8.1.2.1

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

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

Step 8.1.2.2

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

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

Step 8.1.2.3

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

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

Step 8.1.2.4

Cancel the common factor of $2$ .

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Step 8.1.2.4.1

Cancel the common factor.

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

Step 8.1.2.4.2

Rewrite the expression.

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

Step 8.1.2.5

Evaluate the exponent.

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

Step 8.1.3

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

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

Step 8.1.4

Multiply $- \frac{1}{2} \cdot \frac{3}{4}$ .

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Step 8.1.4.1

Multiply $\frac{3}{4}$ by $\frac{1}{2}$ .

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

Step 8.1.4.2

Multiply $4$ by $2$ .

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

Step 8.2

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

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

Step 8.3

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

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Step 8.3.1

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

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

Step 8.3.2

Multiply $2$ by $4$ .

$\frac{4}{8} - \frac{3}{8}$

Step 8.4

Combine the numerators over the common denominator.

$\frac{4 - 3}{8}$

Step 8.5

Subtract $3$ from $4$ .

$\frac{1}{8}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{8}$

Decimal Form:

$0.125$