∫(xe^x+sin (πx/4))dx?

∫(xe^x+sin (πx/4))dx = ∫ x e^x dx + ∫ sin ( πx/4) dx

∫ x e^x dx

Integrate by parts
dv= e^x dx; v= e^x
u = x; du = dx

∫ u dv = u v - ∫ v du
∫ x e^x dx = x e^x - ∫ e^x dx
∫ x e^x dx = x e^x - e^x

∫ sin ( πx/4) dx

Let u= πx/4
du = (π/4) dx
dx = (4/π) du

∫ sin ( πx/4) dx = (4/π) ∫ sin(u) du
= (-4/π) cos(u)
= (-4/π) cos(πx/4)

∫(xe^x+sin (πx/4))dx = x e^x - e^x + (-4/π) cos(πx/4)

= x e^x - e^x - (4/π) cos(πx/4) + C

∫(xe^x+sin (πx/4))dx
= ∫xe^x dx+ int sin (πx/4)dx
= I + J

Integration of parts
I = int xe^x dx

set u = x , du/dx = 1
dv = e^x dx , v = e^x

I = xe^x - int 1.e^x dx
= xe^x - e^x

J = int sin (πx/4)dx
set u = pi x/4
du = pi/4 dx

then J = int sin (u) *4/pi du
= -4cos(u)/pi
= -4 cos(pix/4) /pi

∫(xe^x+sin (πx/4))dx = I + J
= xe^x - e^x -4 cos(pix/4) /pi + c