Hi, I’m currently taking MTH 252 (integral calculus) at my university and I am having an enormous amount of trouble understanding trigonometric integrals and trigonometric substitution. I was wondering if anyone on here could lend a hand or point me in the direction to a webpage that can assist me? I’ve done a bit of searching for websites on this topic but what I have found is in heavy ‘mathspeak’ and is even more confusing to me.
OK - I TA such a class, so I should be able to help.
I presume that you’re au fait with the standard trig identities? The aim of trigonometric substitution is to convert an expression that would be horrendous to integrate otherwise into something relatively trivial.
There’s no hard and fast rule as to what substitution to use where - that comes with practice. But, knowing the derivatives of sin(x), cos(x) and tan(x), is a first step. Secondly, recognising various patterns is useful.
Say if you have to evaluate the integral of (1-x^2)^(-0.5), then recognising that 1-x^2 yields (cos(u))^2, if we make the substitution x=sin(u), makes the integral exceptionally easy, since dx/du is simply cos(u), and terms cancel down, and we end up simply integrating a constant.
So, that’s substitution - does that make sense?
As for trigonometric integration, which bits in particular are you having problems with?
Here are some websites:
http://www.math2.org/math/trig/identities.htm
http://archives.math.utk.edu/visual.calculus/4/substitutions.3/index.html
http://www.math.hmc.edu/calculus/tutorials/trig_substitution/
Thank you so much for the help, the websites were a great help.