Calculus Examples

$\int \frac{\sec^{2} (\frac{1}{x})}{x^{2}} d x$

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

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

$\int - \sec^{2} (u) d u$

Step 2

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

$- \int \sec^{2} (u) d u$

Step 3

Since the derivative of $\tan (u)$ is $\sec^{2} (u)$ , the integral of $\sec^{2} (u)$ is $\tan (u)$ .

$- (\tan (u) + C)$

Step 4

Simplify.

$- \tan (u) + C$

Step 5

Replace all occurrences of $u$ with $\frac{1}{x}$ .

$- \tan (\frac{1}{x}) + C$