# How does one integrate u√(1+u^2) du?

• Rosebud
In summary, the integral of u√(1+u^2) du can be solved by using the substitution method, where the substitution is made to be v=1+u^2. By taking the derivative of v with respect to u, we can find the value of dv to replace the numerator in the original integral. The resulting integral can then be solved using the power rule and the original substitution can be reversed to find the final answer of (1/3)(1+u^2)^(3/2) + C.
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.

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.

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

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.

SammyS said:
Do you know the method of substitution ?
Rosebud said:
No.
So, you do know substitution.

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.

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.

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

## 1. What is the integration of u√(1+u^2) du?

The integration of u√(1+u^2) du is equal to (1/3)(1+u^2)^(3/2) + C.

## 2. How do you solve this type of integration problem?

To solve this type of integration problem, you can use the substitution method. Let u = 1+u^2 and du = 2u du. Then, the integral can be rewritten as (1/2)∫u^(1/2) du, which can be easily integrated.

## 3. Can this integral be solved using other methods?

Yes, this integral can also be solved using integration by parts. Let u = u and dv = √(1+u^2) du. Then, du = du and v = (1/3)(1+u^2)^(3/2). This method may be more complicated but can also lead to the same answer.

## 4. Are there any special cases for this type of integration?

Yes, if the integration limits are from 0 to a, where a is a positive real number, the integral can be simplified to (1/3)(a√(1+a^2) + arcsinh(a)).

## 5. Can this integral be solved without using substitution or integration by parts?

No, this type of integral requires either the substitution method or integration by parts to be solved. Other methods such as partial fractions or trigonometric substitutions may not be applicable.

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