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.
du = sec^2x*dx
I figure that maybe I can "break apart" the sec^3x term into sec^2x*secx, giving me the following:
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
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:
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:
thus h()'=()^1.5 making H()=(()^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.
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