azulita asked in Science & MathematicsMathematics · 1 decade ago

# I need more help with this 1/(g-(k/m)v^2) dv?

I need to make a paper explaining how to solve this integral I need to factor fractions and end with a hyperbolic trig function. I will apreciate any help thanks.

### 1 Answer

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• Dr D
Lv 7
1 decade ago
Favorite Answer

Essentially you want to write the integral as:

(m/k)*dv / [ mg/k - v^2]

And replace mg/k with a^2.

The easiest method is to get the hyperbolic trig function directly from the integration.

Use the substitution

x = a*tanhθ

dx = a*sech^2 θ

a^2 - x^2 = a^2 * (1 - tanh^2 θ) = a^2 * sech^2 θ

So the integral of dx / (a^2 - x^2) becomes

integral of (1/a) dθ

which gives you (1/a) θ = (1/a) * tanh^-1 (x/a)

Now you just need to replace a with sqrt(mg/k) and x with v.

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If you wish to integrate using partial fractions, then convert to hyperbolic trig, check out the links below where I answered very similar questions in the past.

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