Question

how to integrate e^x(1+sinx/1+cosx)?​

Answer 1

Answer:

I'm.. nt.. magician.... just.. human

Answer 2

Step-by-step explanation:

e^x * [(1 + sinx) / (1 + cosx)]

= e^x * [1 + 2sin(x/2) cos(x/2)] / [2cos^2 (x/2)]

= [(1/2) sec^ (x/2) + tan(x/2)] e^x

Let f (x) = tan(x/2)

=> f '(x) = (1/2) sec^2 (x/2)

=> ∫ e^x [(1+sinx)/(1 + cosx)] dx

= ∫ [(1/2) sec^ (x/2) + tan(x/2)] e^x dx

= ∫ [ f (x) + f '(x) ] e^x dx

= f (x) e^x + c

= tan(x/2) e^x + c.