How does one integrate u√(1+u^2) du?

1. Jul 14, 2015

Rosebud

1. The problem statement, all variables and given/known data
How does one integrate u√(1+u^2) du?

2. Relevant equations
Forget what u is being substituted for. It doesn't really matter right now. I just need to integrate what is typed above.

3. The attempt at a solution
I am completely lost and not sure what to do.

2. Jul 14, 2015

SammyS

Staff Emeritus
What's the derivative of 1+u2 ?

3. Jul 14, 2015

Rosebud

The derivative of 1+u^2 with respect to u is 2u du.

I know the answer is (1/3)(1 + u^2)^(3/2) but I don't know how to find it.

4. Jul 14, 2015

SammyS

Staff Emeritus
Do you know the method of substitution ?

5. Jul 14, 2015

Rosebud

No.

6. Jul 14, 2015

SammyS

Staff Emeritus
That should have been a BIG hint.

Take the derivative of (1/3)(1 + u^2)^(3/2) . Then work backwards.

7. Jul 14, 2015

Rosebud

Thanks for the tip. I did that but I still fail to see the connection. Can you, or someone else, give me the next step?

8. Jul 14, 2015

DeldotB

Let v= 1+u^2
so dv=2u du

how do you replace du and (1+u^2)

9. Jul 14, 2015

Rosebud

Numerator? There is no fraction here unless you you consider 1 as the denominator.

10. Jul 14, 2015

DeldotB

ahh, thought I saw a / in front of the root.
fixed it. That partial square root symbol gets me

11. Jul 15, 2015

Rosebud

That helped immensely. Thank you so much. I forgot that I could use substitution more than once.

12. Jul 15, 2015

SammyS

Staff Emeritus
So, you do know substitution.

13. Jul 15, 2015

epenguin

Of course a substitution in a substitution is still one substitution overall so you can substitute twice all at once, but maybe it is better to go by stages at first.

14. Jul 15, 2015

Rosebud

Yes, I know substitution. You asked me if I knew the method of substitution, which I understood as, you asking if I knew which method of substitution that I should use.

15. Jul 16, 2015

OrthoJacobian

1 + [sinh^2(x)] = [cosh^2(x)]