Why does ∫1/√(x²+d²) dx give ln(x+√(x²+d²))?

[physicsprince]

Homework Statement

To find the INTEGRAL of 1/(x^2+d^2)^1/2 integrated with respect to dx.
d is a constant

I tried to write it as :

ln (x^2+ d^2)^1/2

but my book gives an answer of

ln { x + (x^2+ d^2)^1/2 }

i don't understand how. Can you please explain it step by step. Clearly please.

 

[lanedance]

how about starting with the subtitution x = d.sec(u)?

 

[hunt_mat]

Even better, use the substitution:

$$x=d\sinh u$$

 

[lanedance]

actually i meant x = d.tan(u), which makes more sense... but i'd still try huntmat's suggestion

 

[lanedance]

also d is a bad constant to use when you're differentiating as you may get confused, something like a or s would be better