Expert: Abe Mantell - 1/28/2004

Question
I am having problems solving the following definite integraL

The integral from 0 to 2^.25 of:
X^3(4-x^8)^.5 dx
I tried subbing in u for x^4, getting it to the form of (a^2-u^2)^.5 and using trig identities from there, but i keep getting stuck.
any ideas?

Answer
Hello Rebecca,

Yes, good substitution!  Let u=x^4, so du=4x^3 dx
Thus, the integral becomes:
(1/4)INT((2^2-u^2)^(1/2) du) from u=0 to u=2
now let u=2sin(w), so du=2cos(w) dw,
and (2^2-u^2)^(1/2)=(4-4cos^2(w))^(1/2)=2sin(w)
Now the integral becomes:
(1/4)INT(2sin(w)*2cos(w) dw) from w=0 to w=pi/2

Can you take it from here?  You should finally get pi/4

TTYL, Abe