Calculus 2
if we have
∫ (sec² b / tan² b) sec b tan b db
and u = secb
du = secbtanbdb
and we get ∫(u^2)/(u^2-1) du
how do we go from there to :
∫ ( 1 + (1/(u^2-1) du
can somebody explain to me fully why it became that last expression?

A simple way to see what was done is to add and subtract -1 from the numerator:

(provided that , or )

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Suppose you had a slightly more complex integrand, like

How do we know that? (Assume we don't already know the previous result, so that it's not just a matter of multiplying both sides by .) Simple polynomial division:

, and . Subtracting this from gives a remainder of , so

(where and denote "quotient" and "remainder")

replace   with   27   and simplify.

10 , 27 , 10 , 70 , 10 , 35 , 10 , 48

step-by-step explanation: