Chapter 7.5, Problem 19E

To determine

To find the integral without using partial fractions technique.

Answer to Problem 19E

The value of ${\displaystyle \int \frac{2x}{x^2-4}}dx=\ln \left|x^2-4\right|+C$

Explanation of Solution

Given information:

The integral is ${\displaystyle \int \frac{2x}{x^2-4}}dx$

Calculation:

  $\begin{array}{l}{\displaystyle \int \frac{2x}{x^2-4}}dx\\ \text{let }{x}^2-4=t\\ \text{differentiating both side}\\ 2xdx=dt\\ \therefore {\displaystyle \int \frac{2x}{x^2-4}}dx={\displaystyle \int \frac{1}{t}}dt\text{ = }\ln \left|t\right|+C=\ln \left|x^2-4\right|+C\end{array}$

Therefore, the value of ${\displaystyle \int \frac{2x}{x^2-4}}dx=\ln \left|x^2-4\right|+C$