Integrating sqrt(x^2+1)/x^2 using trig

[dtl42]

Homework Statement

$$\int$$$$\frac{\sqrt{x^{2}+1}}{x^{2}} dx$$

Homework Equations

The Attempt at a Solution

I tried a Trig Substitution with x=tan(u) and ended up with $$\int (csc(u))^{2} * sec(u) du$$

From here I am kind of stuck. I tried a few different integration by parts methods, but they got very messy. I also couldn't find any sort of table for this.

 

[tiny-tim]

I tried a Trig Substitution with x=tan(u) and ended up with $$\int (csc(u))^{2} * sec(u) du$$

Hi dtl42!

Yes, you're almost there …

Put cosec^2 = 1 + cot^2, and you have:

$$\int \left(sec(u)\,+\,cot(u)csc(u)\right) du\,.$$ ​

(though personally, I'd have used x = sinhv, giving ∫coth^2(v)dv)

 

[dtl42]

Thanks very much, I can't believe I didn't see that. Its those variations of the Pythag. Identity that get me.