How do you prove the integral formula dxx2+a2=ln(x+x2+a2)+C ?

1 Answer
Sep 11, 2014

Let x=atanθ. dx=asec2θdθ
So, we can write
dxx2+a2=asec2θa2(tan2θ+1)dθ
by tan2θ+1=sec2θ,
=secθdθ=ln|secθ+tanθ|+C
since tanθ=xa and secθ=x2+a2a,
=lnx2+a2+xa+C1
by the log property ln{AB}=lnAlnB,
=lnx2+a2+xln|a|+C1
by setting C=ln|a|+C1,
=lnx+x2+a2+C