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

by Ein Krieger
Tags: integration, parts, proceed
 P: 34 Hey guys, Need you push to proceed further with integration by parts: ∫e3x*3*x2*ydx=y∫e3x*3*x2dx setting u=3*x2-------du=6*x dx dv= e3*xdx--- v= 1/3* e3*x ∫ e3*x*3*x2*ydx=y*(3*x2* 1/3* e3*x-∫6*x*1/3* e3*xdx) =y*(3*x2* 1/3* e3*x-6/3*∫x*e3*xdx) Solving further about x*e3*x u=x---du=dx dv=e3*xdx---v=1/3*e3*x ∫ e3*x*3*x2*ydx=y*(3*x2* 1/3* e3*x-6/3*(x*1/3*e3*x-∫1/3e3*x)
 P: 34 How can we go further with solution as exp(3*x) repeats all the time?
 Mentor P: 20,937 You've done all the hard work. ∫e3xdx is easy, using a simple substitution.
Math
Emeritus
Thanks
PF Gold
P: 38,879

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

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 $x^0= 1$.

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