How does one integrate u√(1+u^2) du?

[Rosebud]

Homework Statement

How does one integrate u√(1+u^2) du?

Homework Equations

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

The Attempt at a Solution

I am completely lost and not sure what to do.

 

[SammyS]

Homework Statement

How does one integrate u√(1+u^2) du?

Homework Equations

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

The Attempt at a Solution

I am completely lost and not sure what to do.

What's the derivative of 1+u² ?

 

[Rosebud]

What's the derivative of 1+u² ?

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.

 

[SammyS]

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.

Do you know the method of substitution ?

 

[Rosebud]

Do you know the method of substitution ?

No.

 

[SammyS]

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.

That should have been a BIG hint.

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

 

[Rosebud]

That should have been a BIG hint.

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

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?

 

[DeldotB]

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

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

 

[Rosebud]

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

Notice you have a u in the numerator. So what can you do to replace the numerator with dv ?

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

 

[DeldotB]

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

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

 

[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.

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

 

[SammyS]

Do you know the method of substitution ?

No.

So, you do know substitution.

 

[epenguin]

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

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.

 

[Rosebud]

So, you do know substitution.

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.

 

[OrthoJacobian]

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