How do you integrate x2x2+1 by trigonometric substitution?

1 Answer

x2dxx2+1=x2x2+112sinh1x+C

Explanation:

x2dxx2+1

After using x=sinhy and dx=coshydy transforms, this integral became

(sinhy)2coshydy(sinhy)2+1

=(sinhy)2coshydy(coshy)2

=(sinhy)2coshydycoshy

=(sinhy)2dy

=cosh2y12dy

=14sinh2yy2+C

=142sinhycoshyy2+C

=12sinhycoshyy2+C

After using x=sinhy, coshy=x2+1 and y=sinh1x inverse transforms, I found

x2dxx2+1=x2x2+112sinh1x+C