Question

How do you prove the integral formula `intdx/(sqrt(x^2+a^2)) = ln(x+sqrt(x^2+a^2))+ C` ?

Answer 1

Let `x=atan theta`. `Rightarrow dx=asec^2theta d theta`
So, we can write
`int{dx}/{sqrt{x^2+a^2}}=int{asec^2theta}/{sqrt{a^2(tan^2theta+1)}}d theta`
by `tan^2theta+1=sec^2theta`,
`=intsec theta d theta=ln|sec theta+tan theta|+C`
since `tan theta=x/a` and `sec theta={sqrt{x^2+a^2}}/a`,
`=ln|{sqrt{x^2+a^2}+x}/a|+C_1`
by the log property `ln{A/B}=lnA-lnB`,
`=ln|sqrt{x^2+a^2}+x|-ln|a|+C_1`
by setting `C=ln|a|+C_1`,
`=ln|x+sqrt{x^2+a^2}|+C`