Calculus Examples

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

Step 1

Let $u = x^{2} - 3 x$ . Then $d u = (2 x - 3) d x$ , so $\frac{1}{2 x - 3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 1.1

Let $u = x^{2} - 3 x$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $x^{2} - 3 x$ .

$\frac{d}{d x} [x^{2} - 3 x]$

Step 1.1.2

Differentiate.

Tap for more steps...

Step 1.1.2.1

By the Sum Rule, the derivative of $x^{2} - 3 x$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x]$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [- 3 x]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 3 x]$ .

Tap for more steps...

Step 1.1.3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$2 x - 3 \frac{d}{d x} [x]$

Step 1.1.3.2

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

$2 x - 3 \cdot 1$

Step 1.1.3.3

Multiply $- 3$ by $1$ .

$2 x - 3$

Step 1.2

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

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

Step 2

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

$\ln (| u |) + C$

Step 3

Replace all occurrences of $u$ with $x^{2} - 3 x$ .

$\ln (| x^{2} - 3 x |) + C$