Integration by Parts:

   1.  Tubular Integration: Tubular Integration is a process when more than one application of parts is needed, this process will speed things up.

ex. 

ò x² sin x dx

First, break the integral as before, u = x³ and dv = cos x dx. Instead of differentiating once, differentiate until 0 is obtained, Also integrate dv repeatedly, then finally, take the products diagonally, appending +,-,+ or - as shown below, the integral is the sum of the products.

ò x² sin x dx = - x cos x + 2x sin x + 2 cos x

Sometimes, when integrating by parts, an integral comes up that is similar to the original one, if that is the case, then this expression can be combined with the original.

ex.

ò e^x cos x dx

e^x sin x - ( - e^x cos x + ò e^x cos x dx )
e^x sin x + e^x cos x - ò e^x cos x dx
ò e^x cos x dx = e^x sin x + e^x cos x - ò e^x cos x dx;

(Add ò e^x cos x dx to both sides:)

2 ò e^x cos x dx = e^x sin x + e^x cos x

The final solution is:

ò e^x cos x dx = ½ (e^x sin x + e^x cos x) + C