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.