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

1 Answer
Jul 25, 2016

f(x)=1/3(x^2-1)sqrt(x^2-1)+3-sqrt3..

Explanation:

Let us subst. x^2-1=t^2 rArr 2xdx=2tdtrArrxdx=tdt.

Hence, intxsqrt(x^2-1)dx=intsqrt(t^2)tdt=intt^2dt=t^3/3

But, x^2-1=t^2 rArr t=(x^2-1)^(1/2)

So, f(x)=intxsqrt(x^2-1)dx=1/3(x^2-1)^(3/2)+C..................(1)

To determine C, we are given the cond. that f(2)=3

By (1), then, we have, 1/3*3^(3/2)+C=3 rArr C=3-sqrt3

Hence, f(x)=1/3*(x^2-1)^(3/2)+3-sqrt3, or,

f(x)=1/3(x^2-1)sqrt(x^2-1)+3-sqrt3..