Integral of (x dx) / (x^2 + a^2)

[hardyaa1]

Integral of (x dx) / (x^2 + a^2) where a is some real constant.

I could look this up in an integral table, but I'd like to actually learn how it's done.

Thanks,
-Aaron

 

[daon]

You have a function which may be written in the form of:

\(\displaystyle \frac{c(u'dx)}{u} = c \frac{du}{u}\)

where c is some real constant.

 

[BigGlenntheHeavy]

\(\displaystyle \int\frac{x}{x^{2}+a^{2}}dx, \ I'm \ assuming \ that \ a \ is \ a \ constant.\)

\(\displaystyle Then \ let \ u \ = \ x^{2}+a^{2}, \ \implies \ \frac{du}{2} \ = \ xdx, \ hence -\)

\(\displaystyle \frac{1}{2}\int\frac{du}{u} \ = \ \frac{ln|u|}{2}+C \ = \ \frac{ln(x^{2}+a^{2})}{2}+C\)

\(\displaystyle Check: \ D_x\bigg[\frac{ln(x^{2}+a^{2})}{2}+C\bigg] \ = \ \frac{1}{2}\bigg[\frac{2x}{x^{2}+a^{2}}\bigg] \ = \ \frac{x}{x^{2}+a^{2}}\)