There are a few circumstances where there are clear "forms" which are appropriate for integrating by parts.
The first of these is when you have some the form int x^n f(x) dx, where f(x) is some function that you can repeatedly integrate. For example, int (x^3sin(x))dx is a prime candidate for integration by parts. You will be repeatedly taking the derivative of x^3 and repeatedly integrating sin(x). After a few application of integration by parts, the x^3 will turn into a 0, giving you a solvable integral.
Another case where integration by parts finds use is when you have the form int f(x)g(x) dx, where after repeated integration by parts, you end up with an integral which resembles your original integral. For example, int (e^xsinx)dx is such a function. The integral of e^x is e^x and after taking the derivative of sinx twice, you end up with another instance of sinx.
We'll solve an example of each.
int (x^2sinx)dx
After applying integration by parts once, we get:
-x^2cosx + int (2xcosx)dx
Apply again to get:
-x^2cosx + 2xsinx -2int(sinx)dx
The integral is now solvable, yielding the answer:
-x^2cosx + 2xsinx + 2cosx + C
Now consider int (e^x sinx)dx.
After applying integration by parts twice, we get:
int(e^xsinx)dx = e^xsinx - e^xcosx - int(e^xsinx)dx
Let I = int(e^xsinx)dx and we have
I = e^xsinx - e^xcosx - I
2I = e^xsinx - e^xcosx
I = (1/2)(e^x)(sinx - cosx)
Since I is our original integral, we've found our answer.
