Integral Homework Help: Solving ∫ dy/√(y²+C) Using ln and sinh^-1 Methods

[Ed Aboud]

Homework Statement

I'm trying to solve $$\int \frac{dy}{\sqrt{y^2 + C}}$$

Homework Equations

The Attempt at a Solution

Is it $$ln |y + \sqrt{y^2 + C^2} |$$

Or is it $$sinh^-^1$$ something.

Thanks.

 

[Dick]

It might be both, your first expression looks rather like sinh^(-1) to me as well. Why don't you try doing it? Put y=C*sinh(u) (assuming you mean y^2+C^2 in the integrand).

 

[Ed Aboud]

It turns out that its $$sinh^-^1 (\frac{y}{c})$$

Because my original differential equation was $$\frac{d^2y}{dt^2} - y = 0$$

Thanks!

 

[Dick]

arcsinh(y/C)=ln(y/C+sqrt(y^2/C^2+1)). That's the same as ln(y+sqrt(y^2+C^2)) up to a constant. They are both right.