integration by parts formula proof year 2

[hidden fjord]

hi, i was watching zeeshans video for recap on integration by parts and i know u rnt supposed to know the proof but for understanding i decided to but when he got to the 3rd step i got confused bc why didnt the highlighted part in yellow cancel out as well?

[uneven smelt]

because v is a function of x so when you multiply it by du/dx and do the integral you won't get v* u

[vivid tangle]

How would he cancel that out? Altho it looks like a fraction you cannot treat it as one.

[hidden fjord]

sorry i dont get it 😭 wdym?

[uneven smelt]

okay think if v was sin(x) and u was x^2 they are both functions of x

[hidden fjord]

um if u differentiate smth and integrate it at the same time dont u basically cancel out both processs?

[hidden fjord]

okay

[uneven smelt]

but it's not right to say that the integral of sin(x) * 2x is the same as sin(x) * x^2 (+c)

[vivid tangle]

I mean yh thats true but notice that its being multiplied by another function of x

[hidden fjord]

where did u get the 2x from?? 😓😓

[hidden fjord]

sorry but why wld that make a difference 😓

[vivid tangle]

A simple way to think of why this isnt the case is that the Integral of 2 Is not the same as integral of 2x right. And likewise the integral os sin(x) is different to that of x sin(x) when you multiply the function by another function of x it changes the integral

[hidden fjord]

ohhh omg

i see that makes sense yes

thank uuuu

[vivid tangle]

Nws