Integrating √(1+x²)/x with trig substitution

[Nathan Wygal]

1. The problem is as follows: ∫(√1+x^2)dx/(x) 2. Using trig sub --> x = atanΘ with a = √1 = 1. So x = tanΘ and dx = sec^2ΘdΘ. 3. Picture included of attempted solution. I tried u substitution with both u = secΘ and u=tanΘ but didn't have the right du. I then tried breaking the sec^3Θ/tanΘ (second to last step shown in work) into sines and cosines but, once again, no luck. Any help would be greatly appreciated.

Note: I hope the format of my question is adequate this time. Sorry for the last post.

 

[Charles Link]

As I previously suggested, try multiplying numerator and denominator by $\sin(\theta)$ and letting $u=\cos(\theta)$ after a little algebra. (e.g. $\sin^2(\theta)=1-\cos^2(\theta))$ Then try using partial fractions to get the integral expression involving "u" in workable form.

 

[vela]

You could also try a hyperbolic trig substititution: $x = \sinh u$.