Please explain how to solve the following integral.

we need to find integral of e^tanx*(sin2x+xsec^2x)

we know that integral of e^tanx*f(x) is e^tanx*(f’(x)+f(x)sec^2x) so if we let f(x)= x then integral of e^tanx*x is e^tanx*(1+xsec^2x)

so write e^tanx*(sin2x+xsec^2x) as e^tanx*(1+xsec^2x) – e^tanx*(1 – sin2x)

obviously we know the integral of e^tanx*(1+xsec^2x) and now need to find e^tanx*(1 – sin2x)

so put y= tanx and use the formula sin2x= 2tanx/(1+tan^2x)

so integral becomes e^y(1 – 2y/(1+y^2))dy/(1+y^2)= e^y(1/(1+y^2) – 2y/(1+y^2)^2)

if f(y)= 1/(1+y^2) then f’(y)= – 2y/(1+y^2)^2 so e^y(1/(1+y^2) – 2y/(1+y^2)^2)= e^y(f(y)+f’(y)) whose integral is known to be e^y*f(y)

combining all these together, we get the final integral as e^tanx*(x – cos^2x) + C