I am struggling to understand why can we just multiply by $dt/dt$. I was thinking it was just a change of variables, but I cannot come up how that works. Can someone explain why we are allowed to do this?
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1$\begingroup$ Also check: en.wikipedia.org/wiki/Integration_by_substitution $\endgroup$– AndrewCommented Dec 1, 2020 at 19:40
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$\begingroup$ Related: physics.stackexchange.com/questions/572956/… $\endgroup$– BrickCommented Dec 1, 2020 at 20:17
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$\begingroup$ No where is there a multiplication by dt/dt. Please clarify $\endgroup$– user196418Commented Dec 5, 2020 at 2:27
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1 Answer
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$x=x(t)$ being a function of time, we can compute it's total differential:
$$dx = \frac{dx}{dt}dt$$ and thus,
$$\int f(x)dx = \int f(x(t))\frac{dx}{dt}dt$$
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$\begingroup$ Thank you for your answer! Now, what if it was a definite integral? lets say from x=1 to x=5, how do we change the limits of integration? $\endgroup$– Ba LaloCommented Dec 1, 2020 at 18:31
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$\begingroup$ you need to invert your equation for $x(t)=1$ to get at which $t$ you have $x=1$. $\endgroup$– fgoudraCommented Dec 1, 2020 at 18:55
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$\begingroup$ that makes a lot of sense! thank you! $\endgroup$– Ba LaloCommented Dec 1, 2020 at 21:13
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$\begingroup$ @BaLalo if this answer suits you don't forget to mark it as the accepted answer ; ) $\endgroup$– fgoudraCommented Dec 2, 2020 at 22:33