# Which integration rule is applied here?

i have a derivation of a physical equation, where there is an equation

$$\int mv \gamma \,\textrm{d}v = \frac{m}{2}\int \gamma \, \textrm{d}(v^2)$$

Q1: How did we derive right side from left one? Could anyone explain this step by step or provide me with names of the integration rules applied here so i can google it myself.

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$$\frac{dv^2}{dv}=2v \Rightarrow \frac{dv^2}2=vdv$$

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Thanks... a simple derivation explains everything. – 71GA Nov 26 '12 at 16:29
In fact sometimes it is easier to me to learn integration through diferentiation. – 71GA Nov 26 '12 at 16:29

This follows directly from

$$\int A f(x)\, \text dx= A\int f(x)\, \text dx$$

and a substitution.

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For what it's worth, I'll mention that $\mbox{d}(v^2)$ is shorthand for $\dfrac{\mbox{d}v^2}{\mbox{d}v}\mbox{d}v$. – Clive Newstead Nov 26 '12 at 15:58
What yo just said is that i can put a constant in front of an integral? TY – 71GA Nov 26 '12 at 16:28
@71GA $m$ is just a constant. Thus you can move it out of the integral. – Argon Nov 26 '12 at 16:29