1. ## indefinite integral problem

I'm trying to find the indefinite integral of:

(root)(x) / (root)(x) - 3 , the -3 is not under the root-sign

I've tried various u-subsitutions, none of which have made the problem any easier. I also tried using polynomial division to simplify the expression to

1 + ( 3 / ((root)(x) - 3), which I still cannot solve.

Would appreciate any help!

2. ## Re: indefinite integral problem

Using the substitution $t=\sqrt{x}$ you'll obtain an easy integral (rational on $t$) .

3. ## Re: indefinite integral problem

Originally Posted by gralla55
I'm trying to find the indefinite integral of:

(root)(x) / (root)(x) - 3 , the -3 is not under the root-sign

I've tried various u-subsitutions, none of which have made the problem any easier. I also tried using polynomial division to simplify the expression to

1 + ( 3 / ((root)(x) - 3), which I still cannot solve.

Would appreciate any help!
Is it $\displaystyle \int{\frac{\sqrt{x}}{\sqrt{x} - 3}\,dx}$? If so...

\displaystyle \begin{align*}\int{\frac{\sqrt{x}}{\sqrt{x} - 3}\,dx} &= \int{\frac{x}{\sqrt{x}(\sqrt{x} - 3)}\,dx} \\ &= 2\int{\frac{x}{2\sqrt{x}(\sqrt{x} - 3)}\,dx}\\ &= 2\int{\frac{u^2}{u - 3}\,du}\textrm{ after making the substitution }u = \sqrt{x} \implies du = \frac{dx}{2\sqrt{x}} \end{align*}

4. ## Re: indefinite integral problem

Thanks! Yes, that is the integral. Actually I tried doing just that, I just didn't know how to evaluate that integral either... Do you use long division to get:

u + (3u / (u +3))

?

Thanks again!

5. ## Re: indefinite integral problem

Originally Posted by gralla55
Thanks! Yes, that is the integral. Actually I tried doing just that, I just didn't know how to evaluate that integral either... Do you use long division to get:

u + (3u / (u +3))

?

Thanks again!
You're on the right track, but it should be $\displaystyle u + \frac{3u}{u - 3}$. You now need to long divide again.

6. ## Re: indefinite integral problem

Lol, you're right, 3u / (u-3) is of course correct. I used long division again and FINALLY got the right answer!

Thank you so much!!