Evaluate the following integrals :\(\int\frac{e^{mtan^{-1}x}}{1+x^2}\)

1.	Evaluate the following integrals :\(\int\frac{e^{mtan^{-1}x}}{1+x^2}\)dx
Answer» Assume, tan^-1x = t d( tan^-1x) = dt ⇒ \(\frac{1}{x^2+1}\) = dt Substituting t and dt ⇒ \(\int\) e^mt dt ⇒ \(\frac{e^{mt}}{m}\)+ c But, t = tan^-1x ⇒ \(\frac{e^{mtan^{-1}x}}{m}\) + c.

Evaluate the following integrals :\(\int\frac{e^{mtan^{-1}x}}{1+x^2}\)dx

Answer»

Assume,

tan^-1x = t

d( tan^-1x) = dt

⇒ \(\frac{1}{x^2+1}\) = dt

Substituting t and dt

⇒ \(\int\) e^mt dt

⇒ \(\frac{e^{mt}}{m}\)+ c

But,

t = tan^-1x

⇒ \(\frac{e^{mtan^{-1}x}}{m}\) + c.