Is Integration by Parts the Key to Solving Complex Equations?

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