On-line Math 21

6.1  Area between curves

Area = 2 ó õ 1 0  æ ç è arctan(x)- px 4 ö ÷ ø dx.

Ah, but how do you do that integral?

ó õ arctan(x)dx

is an integration by parts problem. Take u = arctan(x) , and dv = dx . Then v = x and du = dx/(1+x²). Then,

ó õ arctan(x)dx = x·arctan(x)- ó õ x dx 1+x2 = x·arctan(x)- ó õ xdx 1+x2 .

Now do a substitution. Let's not use the letter u again, though. Substitute w = 1+x² . Then, dw = 2xdx , and it just so happens that we have xdx to deal with, xdx = dw/2 , so

ó õ arctan(x)dx = x·arctan(x)- ó õ xdx 1+x2 = x·arctan(x)- ó õ dw/2 w = x·arctan(x)- 1 2 ln|w|+C = x·arctan(x)- 1 2 ln| 1+x2| +C.

OK, the absolute value signs are not important.

Copyright (c) 2000 by David L. Johnson.