Question about integration by substitution?

[MathWarrior]

I am confused about integration in other cases, I understand that you can use substitution if the derivative exists next to what your trying to integrate then you can use it.

However while studying Arc Length and surface of a revolution I came across a problem such that I had to integrate the following:
integrate from 0 to 8 sqrt(1+x)

Which can be done by substituting u=(1+x) and rewriting it. But I do not see how the derivative exists next to the function in this case? I am sure there are other examples but this is just one I came across. Could someone please enlighten me?

 

[lurflurf]

I would write
sqrt(1+x)dx=2sqrt(1+x)^2[(1/2)/sqrt(1+x)]dx

doing it your way
u=(1+x)
du=dx
which is present in the integrant
sqrt(1+x)dx->sqrt(u)du

 

[dextercioby]

I am confused about integration in other cases, I understand that you can use substitution if the derivative exists next to what your trying to integrate then you can use it.

However while studying Arc Length and surface of a revolution I came across a problem such that I had to integrate the following:
integrate from 0 to 8 sqrt(1+x)

Which can be done by substituting u=(1+x) and rewriting it. But I do not see how the derivative exists next to the function in this case? I am sure there are other examples but this is just one I came across. Could someone please enlighten me?

I don't understand your issue, specifically the part I bolded. What's wrong with the initial function and with the substitution ?
x ---> x(u) (bijective correspondence) is valid for the substitution involved in your problem.

 

[MathWarrior]

I don't understand your issue, specifically the part I bolded. What's wrong with the initial function and with the substitution ?
x ---> x(u) (bijective correspondence) is valid for the substitution involved in your problem.

I just don't see how the derivative exists in this case, I guess, which is why I am confused. My best guess is that its 1 in in this case and 1 is implied to be there.

 

[dextercioby]

Well, they are linearly dependent one of another x=u-1 (u runs in the interval [1,9]) so dx/du = d/du (u-1) = 1.