Integrate acceleration when a = f(v)

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Mentor note: Moved from a technical forum section, so missing the HW template.
Summary:: Integrate acceleration when a = f(v) when separation of variables is not trivial, ie a = k +v^2

When acceleration is a function of velocity, ie there is a friction force, you would separate the variables as such:

dv/dt = v^2

v^-2*dv = dt

and then integrate.
but what do you do when they are not easily separable, as in:

dv/dt = k + v^2

?
 
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Isn't ##\frac 1{k + v^2}## a standard integral?
 
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Note that
a=\frac{dv}{dt}=\frac{dv}{dx}\cdot\frac{dx}{dt}=v\frac{dv}{dx}
Or you can just separate as usual to get:
\frac{1}{k+v^{2}}\frac{dv}{dt}=1
Integrate both sides to get:
\int\frac{dv}{k+v^{2}}=\int dt+C
 
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