Alternative to Integration by Parts?

[Masna]

Hey all!

I was recently refreshing my memory of integration by parts via some personal reading when I thought, there must be a better way. Integration by parts (while creative in that it integrates the entire product rule) feels very arbitrary to what it's attempting to calculate (at least concerning the integral of a product of two functions).

So, I ask, are there any alternatives to integration by parts? Perhaps, any less messy methods? Don't take this the wrong way: integration by parts is brilliant. But I still feel that there has to be a simpler, much more intuitive way to calculate the integral of the product of two functions.

If there aren't any defined alternatives, is there any work being done by any mathematicians (publicly) on a better way to complete said task?

I personally spent a few hours trying to make some connections that might lead me finding an alternative method, I didn't find an alternative method.

Thanks!

 

[pbandjay]

I tend to use the Tabular Method, which is just a "rearranged" way of integration by parts and is a little nicer if you have some type of integral that requires multiple stages of integration by parts, e.g...

$$\int{x^5}{e^x}dx$$

For a more exotic integral that you won't necessarily notice it requires integration by parts, I will revert back to the original way.

Other than the tabular method, I have not seen many more techniques, other than using tables...

 

[Count Iblis]

Differentiation w.r.t. a suitably chosen parameter (or Taylor expanasions w.r.t. that parameter) is often far more efficent than partial integration.

Take e.g. the example given by pbandjay above:

$$\int x^{5}\exp(x)dx$$

The smart way to compute this is by replacing $x^{5}$ by $\exp(px)$. Then, the fifth derivative w.r.t. p at p = 0 will yield the answer. However, computing that fifth derivative will aslo be a bit messy, albeit far less messy that doing a partial integration 5 times. Now, computing high order derivatives is more easily done by obtaining the Taylor expansion and then extracting the derivatives from that.

So, the method works as follows. We have:

$$\int \exp(px)\exp(x)dx = \int \exp\left[(p+1)x\right]dx=\frac{\exp\left[(p+1)x\right]}{p+1}=\exp(x)\frac{\exp(px)}{1+p}$$

If you expand the factor

$$\frac{\exp(px)}{1+p}$$

in powers of p by multiplying the Taylor series of thegeometric series and of the exponential function and extract the coefficient of p^5, you easily find that it is given by:

$$\frac{x^5}{5!}-\frac{x^4}{4!}+\frac{x^3}{3!}-\frac{x^2}{2!}+x-1$$

The coeficient of p^5 of the integral

$$\int \exp(px)\exp(x)dx$$

is 1/5! times the desired integral, so we find:

$$\int x^{5}\exp(x)dx = \left(x^5-5x^4+20x^3-60x^2+120x-120\right)\exp(x) + c$$