Answer to Question #5318 in Calculus for Akhtar Rasool khan

Question

sir please solve"Integral e^x(tanx+1)/secxdx.
best regards

Answer 1

it's known that secx=1/cos(x), tan x=sinx/cosx
hence (tanx+1)/secx=(tanx+1)cosx=(sinx+cosx)
So
Integral(e^x(tanx+1)/secx)dx=Integral(e^x*(sinx+cosx))dx=Integral(e^x*sinx)dx+Integral(e^x*cosx)dx (*)
Consider the first one:
Integral(e^x*sinx)dx = e^x*sinx-Integral(e^x*cosx)dx +const (**)
using integration by parts u=sinx => u'=cosx
v'=e^x => v=e^x
Substituting obtained result (**) into formula for initial integral (*) we get
Integral(e^x(tanx+1)/secx)dx=e^x*sinx-Integral(e^x*cosx)dx +const+Integral(e^x*cosx)dx=e^x*sinx+const