Integrating a Definite Integral with an Undefined Function at One Endpoint

[Aceix]

Homework Statement

integrate from 1 to 2 x(x^2-3)^(1/2) with respect to x.

Homework Equations

The Attempt at a Solution

i attempted using numerical approximations but at x=1, the function is not defined so is there a way to combine improper integrals with this?

Aceix.

 

[fourier jr]

I read your problem as $$\int_{1}^{2} x\sqrt{x^{2} - 3}\,dx$$ & used the substitution $u = x^{2} - 3$ then $\frac{1}{2}du = x\,dx$ so it's no improper integral. I'm not sure why you say the function is not defined at x=1.

 

[Dick]

I read your problem as $$\int_{1}^{2} x\sqrt{x^{2} - 3}\,dx$$ & used the substitution $u = x^{2} - 3$ then $\frac{1}{2}du = x\,dx$ so it's no improper integral. I'm not sure why you say the function is not defined at x=1.

It's not defined because at x=1 because 1-3 is negative and square root of a negative is not defined. You would have to properly define the square root as a complex number to be able to integrate.

 

[Aceix]

It's not defined because at x=1 because 1-3 is negative and square root of a negative is not defined. You would have to properly define the square root as a complex number to be able to integrate.

So how do I define the square root as a complex number?

 

[fourier jr]

argh sorry about that but I did notice that when I tried $\int^{2}_{1} \frac{x\,dx}{\sqrt{x^2 - 3}}$ just in case I misread the original post

 

[Dick]

So how do I define the square root as a complex number?

If $x^2-3$ is negative then the complex square roots are either $i \sqrt{3-x^2}$ or $-i \sqrt{3-x^2}$. You have to pick which one you want. This is called 'choosing a branch'. Why are you doing this problem?

 

[Aceix]

If $x^2-3$ is negative then the complex square roots are either $i \sqrt{3-x^2}$ or $-i \sqrt{3-x^2}$. You have to pick which one you want. This is called 'choosing a branch'. Why are you doing this problem?

Saw it in a book(preparing for an exam).

 

[Dick]

Saw it in a book(preparing for an exam).

The problem has two possible answers since you need to make a branch choice. If you aren't really doing complex numbers, then possibly i) they just expect you to say it's not defined or ii) it's a typo.

 

[Aceix]

The problem has two possible answers since you need to make a branch choice. If you aren't really doing complex numbers, then possibly i) they just expect you to say it's not defined or ii) it's a typo.

thanks!