What is f(x) = int xsqrt(x^2-1) dx if f(3) = 0 ?

1 Answer
Aug 2, 2016

f(x) = 1/3(x^2-1)^(3/2) - (16 sqrt 2)/(3)

Explanation:

int xsqrt(x^2-1) dx

= int \ d/dx ( 2/3* 1/2(x^2-1)^(3/2) ) dx

= int \ d/dx ( 1/3(x^2-1)^(3/2) ) dx

= 1/3(x^2-1)^(3/2) + C

using the IV:

0 = 1/3(3^2-1)^(3/2) + C implies C = - (16 sqrt 2)/(3)

So f(x) = 1/3(x^2-1)^(3/2) - (16 sqrt 2)/(3)