Calculus Examples

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

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.

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

Step 2

Simplify.

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Step 2.1

Move the negative in front of the fraction.

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

Step 2.2

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

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

Step 2.3

Move $2$ to the left of ${u_{2}}^{2}$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Apply basic rules of exponents.

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Multiply $2$ by $- 1$ .

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

Step 6

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

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

Step 7

Substitute and simplify.

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Step 7.1

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

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

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.

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

Step 7.2.2

Multiply $- 1$ by $2$ .

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

Step 7.2.3

One to any power is one.

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

Step 7.2.4

Add $- 2$ and $1$ .

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

Step 7.2.5

Multiply $- 1$ by $- 1$ .

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

Step 7.2.6

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

$\frac{1}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$