Integrate x/(1+x): Step-by-Step Guide

[jasmaster35]

How to integrate: x / (1+x)

 

[cepheid]

"By parts" would be my first stab at it

 

[FrogPad]

Would integrating, $$\frac{1}{1+x}$$ be any easier?

If you got rid of the $x$ on the top it might make your life easier right?

$$\frac{x}{1+x}$$

$$\frac{x}{1+x} - \frac{1+x}{1+x} = \frac{-1}{1+x}$$

$$\frac{x}{1+x}=\frac{-1}{1+x}+\frac{1+x}{1+x}$$

$$\frac{x}{1+x} = \frac{1+x}{1+x} - \frac{1}{1+x}$$

$$\frac{x}{1+x} = 1 - \frac{1}{1+x}$$

Obviously you don't have to do that in as many steps.

 

[radou]

How to integrate: x / (1+x)

Use the substitution 1 + x = t.