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 $x^0= 1$.

 

[CellCoree]

dx)
=y*(3*x2* 1/3* e3*x-2/9*e3*x-∫2/9*e3*x)
=y*(x2* e3*x-2/9*e3*x2-∫2/9*e3*x2)
=y*(x2* e3*x-2/9*e3*x2-2/9*e3*x3) + C

Integration by parts is a powerful tool in solving complex equations, but it is not always the key to solving them. In some cases, other techniques such as substitution or partial fractions may be more effective. However, integration by parts is a useful method to have in your toolbox, as it allows you to break down an integral into simpler parts and apply different rules to each part. This can be especially helpful when dealing with integrals that involve products of functions, as shown in the example above. With practice and familiarity, integration by parts can be a valuable tool in solving complex equations.