How does one integrate u√(1+u^2) du?

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Homework Help Overview

The discussion revolves around the integration of the expression u√(1+u^2) du. Participants are exploring methods to approach this integral, with a focus on substitution techniques.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants express confusion about how to proceed with the integration, with some questioning the derivative of the expression 1+u^2. Others suggest using substitution and discuss the implications of the derivative in relation to the integral.

Discussion Status

There is an ongoing exploration of substitution methods, with some participants providing hints and tips. While some guidance has been offered, there is no explicit consensus on the next steps, and participants continue to seek clarification and further assistance.

Contextual Notes

Some participants indicate a lack of familiarity with substitution methods, while others reflect on the potential for multiple substitutions within the integration process. There is also mention of confusion regarding the notation used in the integral.

Rosebud
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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.
 
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Rosebud said:

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 ?
 
SammyS said:
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.
 
Rosebud said:
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 ?
 
SammyS said:
Do you know the method of substitution ?
No.
 
Rosebud said:
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.
 
SammyS said:
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?
 
Let v= 1+u^2
so dv=2u du

how do you replace du and (1+u^2)
 
DeldotB said:
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.
 
  • #10
Rosebud said:
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
 
  • #11
Rosebud said:
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.
 
  • #12
SammyS said:
Do you know the method of substitution ?
Rosebud said:
No.
So, you do know substitution.
 
  • #13
Rosebud said:
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. :oldsmile:
 
  • #14
SammyS said:
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.
 
  • #15
1 + [sinh^2(x)] = [cosh^2(x)]
 

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