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Answer on Calculus Question for Harjyot Singh

Question #10294
Sir,
I'm having a concept problem and would be glad if you can help me out.
my problem is as follows -
1. l dx = x + c right? What does this actually mean? does it mean we are integrating a very small x over and over so we get the parent function?

2. l [v(0), v(t)] dv = v(t) - v(0) ,how? here v is basically a f(t) ,thus to integrate, doesn't it mean
l[0,t] df(t) = f(t) - f(0)?

3. | dv = v, not v+ c, why? where v is velocity
please It would help me a lot if you can clear my concept.
Expert's answer
1. From the definition of the parent function: F is parent to f iff F'=f. For any number c: F'=(F+c)', so we can say that f have infinitely many parent functions which generaly expressed by F+c.Now (x+c)'=1, so I(1dx)=x+c
2. Here f(t)=1. If we rewrite the integral I [v(0),v(t)] dv=I [0,t] v'dt=v[0,t]=v(t)-v(0). See the respective formula in lections or
wherever.
3. Here we think of it as follows I dv=I a dt=v where a is an acceleration. We usually talk about some specfiic acceleration, which is a derivative of a
velocity of some defined physical object. So there is not infinitely many
velocities (v+c, c is any) but one, defined v.

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