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I have an equation like this

$dW=x\,dx$ , where W is work.

I have find a lot on internet, but I wasn't able to find clear information. I have two questions:

1) If I want to integrate this equation, do im forced to apply definite integral or indefinite integral at the same time to each member of the equation ?

2)The right part of the integral is quite clear to me, for example I can integrate between $x=0$ and $x=5$. But in the left part, what's the meaning to integrate between (for example) $W=4$ and $W=8$ ?

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If you are finding the work between two points you can integrate the right hand side from between any values of $x$.
and the left hand side between any values of $x$ as W is a function of$ x$.
That being said , if you are integrating from$ 2 $to $x $ you'll need to know the initial condition of $W(2)$.
$W(x) - W(2)=\frac{1}{2}x^2 -2$.
For between 2 and x. But notice the initial condition of work at x=0 will be zero.

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