Question

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

Answer 1

`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`..