On-line Math 21

5.4  Techniques of integration

Example 5

ó õ x2sin(x)dx.

Solution

This one needs two applications of the trick. First, set u = x² and dv = sin(x) dx . Then you can easily see that du = 2x dx and v = -cos(x) . Applying parts gives

ó õ x2sin(x)dx = -x2 cos(x)+ ó õ 2xcos(x) dx.

This doesn't finish the problem, but you apply parts the second time to the left-over integral, and it will. This time, u = x (the 2 can be taken out of the integral) and dv = cos(x) dx , and so du = dx and v = sin(x) . Then,

2 ó õ xcos(x) dx = 2x sin(x)-2 ó õ sin(x) dx.

Finally,

ó õ x2sinxdx = -x2 cosx+2x sinx+2cosx+C.

Copyright (c) 2000 by David L. Johnson.