How and when should I integrate by parts?

Analyse the integral, common functions in which integration by parts is the best method include lnx and polynomials of x multiplied by exponential functions or sine and cosine functions. If this is the case with this function split the function up in to two and label the polynomial part u and the other dv/dx.

Find du/dx and v.

sub u, v and du/dx into the following equation:

uv - integral(v du/dx)

Analyse the next integral for the best method to solve, it may require integration by parts again.

We label the polynomial u to be differentiated because when it is subbed into the final equation the order of the polynomial drops. If this polynomial began with an order of 1, such as 2x, you would end up with a constant multiplied by either the exponential or sine/cosine function inside the second integral which could now be solved much easier than before.