Find solution to exercise:What is integral of (cos x)*(e^2x)?

let f(x) = cosx*e^2x

We need
to find the integral of f(x).

==> intg f(x) = intg
cosx * e^2x dx

We will use partial integration to
solve.

Let us assume that:

u=
e^2x   ==>    du = 2e^2x dx

dv = cosx dx 
==>    v = sinx

==> intg udv = u*v - intg
vdu

                     = sinx*e^2x + 2 intg sinx*e^2x
dx.............(1)

Now we will apply the rule
again.

Let u = e^2x  ==>   du
2e^2x

      dv = sinx dx  ==>    v =
-cosx

==> intg sinx*e^2x dx = -cosx*e^2x +2 intg
cosx*e^2x dx

==> But we know that intg cosx*e^2x dx
= intg udv

==> intg udv = sinx*e^2x + 2[
cosx*e^2x -2intg udv]

==> intg udv = sinx*e^2x +
2cosx*e^2x -4intg udv.

==> 5intg udv =
sinx*e^2x +2cosx*e^2x

==> intg udv =
e^2x ( sinx+2cosx) / 5