Concept problem in integration with u-substitution

AjitZK

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Question:

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Solution:

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I was working on question 10a trying to get the first required proof. However, I couldn't do it and checked the solution. The red arrow I have drawn on the solution shows what I have difficulty in understanding. Why can (k-u) become (k-x) when u=k-x? Thank you in advance and have a good day.
 

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Question: ...
Solution:

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I was working on question 10a trying to get the first required proof. However, I couldn't do it and checked the solution. The red arrow I have drawn on the solution shows what I have difficulty in understanding. Why can (k-u) become (k-x) when u=k-x? Thank you in advance and have a good day.

The short answer is they are dummy variables so you can call them anything you want to. That is
\(\displaystyle \int_a^b\, f(x)\, dx\, =\, \int_a^b\, f(u)\, du\, =\,\int_a^b\, f(t)\, dt\, =\,\int_a^b\, f(v)\, dv\, =\, ...\)

The longer answer is to again make another substitution letting (a different) x=u so dx=du; when x=0, u=0; ...
 
Thank you!

So, as it is only a dummy variable, it can be used like that. I understand in that case. Thank you!
 
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