Advanced Integration-Tabular Method

[BurgooKing]

"Advanced" Integration-Tabular Method

When cannot I not use this method?
If the integral is cyclic is there a way to get around it?
Any other information would be nice
Thanks

 

[lurflurf]

Tabular integration is just a systematic way to integrate by parts multiple times. If the original integral reappears (without all other terms having canceled) you can stop and solve for it. A useful case is if p(x)f(x) where f is easy to integrate and p(x) is a polynomial so it will become zero after some number of differentiations.

Usual examples include

$$\int e^{-s t}\cos(a t) \text{ dt}$$
$$\int (x^2+3x+1)\sin(t) \text{ dt}$$
$$\int t^7 e^{-t} \text{ dt}$$

 

[lugita15]

Note that you should be careful when the original integral reappears. Try integrating 1/x using integration by parts, with u = 1/x and dv=dx