week THREE

This week was ALL about Trig. Sub.

Trig. Sub. is used anytime there is a square root in the form of sqrt(a^2 - x^2), sqrt (x^2 - a^2), or sqrt(x^2 + a^2) in an integration problem.

If you see sqrt(a^2 - x^2) then x=asin(theta) and sqrt(a^2 - x^2)=acos(theta).

If you see sqrt (x^2 - a^2) then x=asec(theta) and sqrt (x^2 - a^2)= atan(theta).

If you see sqrt(x^2 + a^2) then x=atan(theta) and sqrt(x^2 + a^2)=asec(theta).

After figuring out which one you have to use, you then find dx by taking the derivative of x and if you have an (x) variable in the problem, then solve for that using the x equation.

After integrating, you have to switch the theta versions of the problem back to x.

Some issues I'm having with this is when you actually have to integrate.
So Ryan, here's some integration formulas to remember:
Ssec = lnsec + tan
Ssec^2x = tan
Ssec^3x = (1/2)sectan + (1/2)lnsec + tan
Scsc*cot = -csc
Stan = -lncos
Scot = lnsin

And some trigonometry formulas to remember:
sin^2x + cos^2x = 1
1 + tan^2x = sec^2x
1 + cot^2x = csc^2x
sin2x = 2sinxcos

Ryan