Why is this legal? (limits of integration)

[iScience]

i have one expression, and then i will expand that expression into another expression like so...

y=f(x) ------> f(x)= x+5u , u= another variable

now i will integrate both sides.. like so...

∫f(x)dx=∫(x+5u)du

now, one fundamental rule in algebra is that, whatever is done to one side, must be done to the other side as well. so i understand that if i integrate something by one side, i must integrate the other side as well, ie i understand why doing something to both sides will make them remain equal to each other. But what i don't get, is why is it legal to integrate one side by one variable and integrate the other side by another variable. I don't understand what makes them remain equal to each other when one does this.

furthermore, even if we were to integrate both sides by the same variable, i don't understand why it is legal to have different limits of integration on both sides. that is.. i don't understand what makes the two expressions remain equal to each other when i integrate one side with one pair of limits, and i integrate the other side with a different pair of limits.

 

[SteamKing]

You are not alone. I don't understand what you have written, either.

In your example, you have two variables, x and u. What is the relationship between x and u? Is u another function of x? Is x another function of u? Why didn't you write your integral of (x + 5u) with respect to x?

With definite integrals, the choice of limits of integration always depends on the variable being integrated. If there are different variables on the LHS and RHS, then there will be different limits of integration.