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Serggg [28]
3 years ago
7

Calculus 2

Mathematics
1 answer:
krek1111 [17]3 years ago
5 0

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

\dfrac{u^2-1+1}{u^2-1}=\dfrac{u^2-1}{u^2-1}+\dfrac1{u^2-1}=1+\dfrac1{u^2-1}

(provided that u^2-1\neq0, or u\neq\pm1)

###

Suppose you had a slightly more complex integrand, like

\dfrac{u^3}{u^2-1}=u+\dfrac u{u^2-1}

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 u.) Simple polynomial division:

u^3=\boxed{u}_{\,q}\cdot u^2, and u(u^2-1)=u^3-u. Subtracting this from u^3 gives a remainder of u^3-(u^3-u)=\boxed{u}_{\,r}, so

\dfrac{u^3}{u^2-1}=\boxed{u}_{\,q}+\dfrac{\boxed{u}_{\,r}}{u^2-1}

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

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