Partial Fractions, Irreducible quadratic factors

[rocomath]

Arc Length, Irreducible quadratic factors

i'm having a hard time seeing this method, and i have to use this method on one of the problems I'm doing to find it's Arc Length.

$$L=\int_{\sqrt{2}}^{\sqrt{1+e^{2}}}\frac{v^{2}dv}{v^{2}-1}}$$

the book suggests to first divide then use a u-substitution. i know that when the power in the numerator is greater than the denominator, i can perform long-division, but i don't see how i can divide this.

 

[sutupidmath]

i don't know whether i am gettin u well, but you could do this inegral by doing this to the function under the integral sign

v^2/(v^2-) =(v^2-1+1)/(v^2-1) = (1- 1/(v-1)(v+1) )

then you could use partial fraction by letting

1/(v-1)(v+1) = A/(v-1) + B /(v+1), and then finding the values of B and A. Pardone me if i misunderstood u.

 

[Gib Z]

Well, like sutupidmath did, add one and then subtract one to the numerator. Then, instead of factoring the new denominator and partial fractions-ing, Try a hyperbolic substitution, I think you will find it quicker.