Going further with integration by parts. Don't know whether to proceed further or not

Ein Krieger

Hey guys,

Need you push to proceed further with integration by parts:

∫e^3x*3*x²*ydx=y∫e^3x*3*x²dx

setting u=3*x²-------du=6*x dx
dv= e^3*xdx--- v= 1/3* e^3*x

∫ e^3*x*3*x²*ydx=y*(3*x²* 1/3* e^3*x-∫6*x*1/3* e^3*xdx)
=y*(3*x²* 1/3* e^3*x-6/3*∫x*e^3*xdx)
Solving further about x*e^3*x
u=x---du=dx
dv=e^3*xdx---v=1/3*e^3*x
∫ e^3*x*3*x²*ydx=y*(3*x²* 1/3* e^3*x-6/3*(x*1/3*e^3*x-∫1/3e^3*x)

Ein Krieger

How can we go further with solution as exp(3*x) repeats all the time?

Mark44

You've done all the hard work. ∫e^3xdx is easy, using a simple substitution.

HallsofIvy

If you have [itex]\int x^n f(x)dx[/itex], where "f" is easy to integrate any number of times (and the "nth" integral of [itex]e^{3x}[/itex] is [itex](1/3^n)e^{3x}[/itex]), just continue taking [itex]u= x^n[/itex], [itex]dv= f(x)dx[/itex]. Everytime du will have x to a lower power until, eventually, it is just [itex]x^0= 1[/itex].