Hi
I'm working on a second order homogeneous ODE that involves integrating
\(\displaystyle \int \frac{1}{\sqrt{1+v^{2}}} \: dv \)..........(see attachment)
I would have made the substitution v = tan(theta) for the integrand sec(theta) wrt theta but my lecturer goes from the integral to an answer of arcsinh(v)
Why/how have they gotten this answer and how do I know to make this substitution in the future?
Thanks
I'm working on a second order homogeneous ODE that involves integrating
\(\displaystyle \int \frac{1}{\sqrt{1+v^{2}}} \: dv \)..........(see attachment)
I would have made the substitution v = tan(theta) for the integrand sec(theta) wrt theta but my lecturer goes from the integral to an answer of arcsinh(v)
Why/how have they gotten this answer and how do I know to make this substitution in the future?
Thanks
Attachments

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