Integrating Rational Functions with Trigonometric Substitutions

[fermio]

How to integrate integral
$$\int\frac{8}{x^2+4}dx$$
?

 

[arildno]

Rewrite it in the form:
$$\int\frac{2dx}{(\frac{x}{2})^{2}+1}$$
See if that helps you.

 

[HallsofIvy]

The derivative of arctan(x) is
$$\frac{d tan^{-1}(x)}{dx}= \frac{1}{x^2+ 1}$$

Does that help?

 

[fermio]

$$\int\frac{8}{x^2+4}dx=\int\frac{2dx}{(\frac{x}{2})^2+1}=\int\frac{4d(\frac{x}{2})}{(\frac{x}{2})^2+1}=4\arctan\frac{x}{2}$$
$$d(\frac{x}{2})=\frac{1}{2}dx$$
$$dx=2d(\frac{x}{2})$$