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
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