Calculus Examples

$\int_{\frac{6}{π}}^{\frac{2}{3 π}} \frac{\cos (\frac{1}{x})}{x^{2}} d x$

Step 1

Let $u = \frac{1}{x}$ . Then $d u = - \frac{1}{x^{2}} d x$ , so $- d u = \frac{1}{x^{2}} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = \frac{1}{x}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $\frac{1}{x}$ .

$\frac{d}{d x} [\frac{1}{x}]$

Step 1.1.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$\frac{d}{d x} [x^{- 1}]$

Step 1.1.3

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

$- x^{- 2}$

Step 1.1.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{x^{2}}$

Step 1.2

Substitute the lower limit in for $x$ in $u = \frac{1}{x}$ .

$u_{lower} = \frac{1}{\frac{6}{π}}$

Step 1.3

Simplify.

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Step 1.3.1

Multiply the numerator by the reciprocal of the denominator.

$u_{lower} = 1 \frac{π}{6}$

Step 1.3.2

Multiply $\frac{π}{6}$ by $1$ .

$u_{lower} = \frac{π}{6}$

Step 1.4

Substitute the upper limit in for $x$ in $u = \frac{1}{x}$ .

$u_{upper} = \frac{1}{\frac{2}{3 π}}$

Step 1.5

Simplify.

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Step 1.5.1

Multiply the numerator by the reciprocal of the denominator.

$u_{upper} = 1 \frac{3 π}{2}$

Step 1.5.2

Multiply $\frac{3 π}{2}$ by $1$ .

$u_{upper} = \frac{3 π}{2}$

Step 1.6

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

$u_{lower} = \frac{π}{6}$

$u_{upper} = \frac{3 π}{2}$

Step 1.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{\frac{π}{6}}^{\frac{3 π}{2}} - \cos (u) d u$

Step 2

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$- \int_{\frac{π}{6}}^{\frac{3 π}{2}} \cos (u) d u$

Step 3

The integral of $\cos (u)$ with respect to $u$ is $\sin (u)$ .

$- (\sin (u)]_{\frac{π}{6}}^{\frac{3 π}{2}})$

Step 4

Evaluate $\sin (u)$ at $\frac{3 π}{2}$ and at $\frac{π}{6}$ .

$- ((\sin (\frac{3 π}{2})) - \sin (\frac{π}{6}))$

Step 5

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$- (\sin (\frac{3 π}{2}) - \frac{1}{2})$

Step 6

Simplify.

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Step 6.1

Simplify each term.

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Step 6.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$- (- \sin (\frac{π}{2}) - \frac{1}{2})$

Step 6.1.2

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

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

Step 6.1.3

Multiply $- 1$ by $1$ .

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

Combine the numerators over the common denominator.

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

Step 6.5

Simplify the numerator.

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Step 6.5.1

Multiply $- 1$ by $2$ .

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

Step 6.5.2

Subtract $1$ from $- 2$ .

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

Step 6.6

Move the negative in front of the fraction.

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

Step 6.7

Multiply $- - \frac{3}{2}$ .

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Step 6.7.1

Multiply $- 1$ by $- 1$ .

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

Step 6.7.2

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

$\frac{3}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{2}$

Decimal Form:

$1.5$

Mixed Number Form:

$1 \frac{1}{2}$