Expert: Socrates - 9/17/2006

Question
definite integral from 0 - pi/3

tanxsec^3/2dx

if you can just get me started, that'd be great.  

i can do most powers and products of trig functions, but the 3/2 power on the sec scares me.  i have no idea what to do with this.  

thanks in advance

Answer
You want S (tanx)(secx)^3/2 dx

Make the substitution u = secx

Then du/dx = (secx)(tanx)

dx = du / (secx)(tanx)

This makes the integral


S (tanx)(secx)^3/2 du / (secx)(tanx) =

S (secx)^1/2 du =

S u^1/2 du =

(2/3)u^(3/2)

The limits for u will be u = sec(0) and u = sec(pi/3)

so the new limits for the variable u are from 1 to 2

the answer will thus be

(2/3)(2^(3/2)) - (2/3)(1^(3/2)) =

(2/3)(2^(3/2) - 1)