Calculus Examples

$\int \frac{\sin (\frac{5}{x})}{x^{2}} d x$

Step 1

Let $u = \frac{5}{x}$ . Then $d u = - \frac{5}{x^{2}} d x$ , so $1 d u = - \frac{5}{x^{2}} d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 1.1

Let $u = \frac{5}{x}$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $\frac{5}{x}$ .

$\frac{d}{d x} [\frac{5}{x}]$

Step 1.1.2

Since $5$ is constant with respect to $x$ , the derivative of $\frac{5}{x}$ with respect to $x$ is $5 \frac{d}{d x} [\frac{1}{x}]$ .

$5 \frac{d}{d x} [\frac{1}{x}]$

Step 1.1.3

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

$5 \frac{d}{d x} [x^{- 1}]$

Step 1.1.4

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

$5 (- x^{- 2})$

Step 1.1.5

Multiply $- 1$ by $5$ .

$- 5 x^{- 2}$

Step 1.1.6

Simplify.

Tap for more steps...

Step 1.1.6.1

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

$- 5 \frac{1}{x^{2}}$

Step 1.1.6.2

Combine terms.

Tap for more steps...

Step 1.1.6.2.1

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

$\frac{- 5}{x^{2}}$

Step 1.1.6.2.2

Move the negative in front of the fraction.

$- \frac{5}{x^{2}}$

Step 1.2

Rewrite the problem using $u$ and $d u$ .

$\int - \frac{1}{5} \sin (u) d u$

Step 2

Since $- \frac{1}{5}$ is constant with respect to $u$ , move $- \frac{1}{5}$ out of the integral.

$- \frac{1}{5} \int \sin (u) d u$

Step 3

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

$- \frac{1}{5} (- \cos (u) + C)$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Simplify.

$- \frac{1}{5} (- \cos (u)) + C$

Step 4.2

Simplify.

Tap for more steps...

Step 4.2.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{5}) \cos (u) + C$

Step 4.2.2

Multiply $\frac{1}{5}$ by $1$ .

$\frac{1}{5} \cos (u) + C$

Step 4.2.3

Combine $\frac{1}{5}$ and $\cos (u)$ .

$\frac{\cos (u)}{5} + C$

Step 5

Replace all occurrences of $u$ with $\frac{5}{x}$ .

$\frac{\cos (\frac{5}{x})}{5} + C$

Step 6

Reorder terms.

$\frac{1}{5} \cos (\frac{5}{x}) + C$