u substitution problem  integration
Find the indefinite integral for sec^3x*tanx using u substitution.
I've done pretty well solving u substitution problems up to this point, but this problem continues to stump me; there is something I am missing. let u=tanx du = sec^2x*dx I figure that maybe I can "break apart" the sec^3x term into sec^2x*secx, giving me the following: sec^2x*secx*u*dx But du=sec^2dx so substituting gives secx*u*du But I can't find the anti derivative of this :( 
There was a time when I knew this stuff. Sadly I've forgotten it all. My mind is tuned with boolean algebra and logic circuits ATM. Sorry...

assuming secx*u*du is correct, the integral to u would be
0.5*secx*u^2+C To see why, just reverse it. Again, not really a math wiz in the formal sense, so please check yourself. Also, this has been ages. 
Maybe a better way is to write:
sqrt(1+tan(x)^2)^3*tan(x) dx Because then if u=tan(x) you get: sqrt(1+u^2)^3*u du > ((1+u^2)^0.5)^3*u du > (1+u^2)^1.5*u du which eliminates the extra x, which I am not sure what to do about (integrate the result to x aftewards?). Finding the anitderivative should be easy from there on, but is has been ages, so you best do that yourself 
Just a thought:
(1+u^2)^1.5*u du looks like h(g(u))'*g(u)' with g(u)=1+u^2 thus h()'=()^1.5 making H()=(()^2.5)/2.5 ((1+u^2)^2.5)/2.5 However, the derivative of 1+u^2 is 2u so apparently there was a separate multiplicative term which ended up as 1/u. Then try to compensate for this. 
All times are GMT 6. The time now is 02:59 PM. 