integration by parts

Question

integral(3x+5)e^2x

Answer 1

Let S represent integral of.

S((3x+5)e^2xdx)=3S(xe^2xdx)+5S(e^2xdx)=3S(xe^2xdx)+(5/2)e^2x.

Let u=x and dv=e^2xdx; du=dx and v=(1/2)e^2x;

uv=(1/2)xe^2x; d(uv)=vdu+udv=(1/2)e^2xdx+xe^2xdx. Applying integrals:

(1/2)xe^2x=(1/2)e^2x+S(xe^2xdx); S(xe^2xdx)=(1/2)xe^2x-(1/2)e^2x.

S((3x+5)e^2xdx)=(3/2)e^2x(x-1)+(5/2)e^2x=(e^2x)(3x/2+1).