Calculus Examples

$\int \frac{2}{x - 2} d x$

Step 1

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

$2 \int \frac{1}{x - 2} d x$

Step 2

Let $u = x - 2$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = x - 2$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $x - 2$ .

$\frac{d}{d x} [x - 2]$

Step 2.1.2

By the Sum Rule, the derivative of $x - 2$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 2]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 2]$

Step 2.1.3

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

$1 + \frac{d}{d x} [- 2]$

Step 2.1.4

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$1 + 0$

Step 2.1.5

Add $1$ and $0$ .

$1$

Step 2.2

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

$2 \int \frac{1}{u} d u$

Step 3

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$2 (\ln (| u |) + C)$

Step 4

Simplify.

$2 \ln (| u |) + C$

Step 5

Replace all occurrences of $u$ with $x - 2$ .

$2 \ln (| x - 2 |) + C$