When to integrate using a hyperbolic substitution instead of normal trig

Lula

New member
Joined
Jan 11, 2021
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6
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
 

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HallsofIvy

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Jan 27, 2012
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It really doesn't matter, use whichever is simplest! Given "\(\displaystyle 1+ v^2\)" you can think "\(\displaystyle u^2= 1+ v^2\) so that \(\displaystyle u^2- v^2= 1\) and match that to either \(\displaystyle tan^2(\theta)- sec^2(theta)= 1\) or \(\displaystyle cosh^2(x)- sinh^2(x)= 1\).

Many people prefer to stick with "sine and cosine" or "sinh and cosh" rather than going to tangent, secant, etc.
 

Subhotosh Khan

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Jun 18, 2007
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23,779
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
One difference I remember between sin(x) and sinh(x) is that:

sin(x) is oscillating function with restricted range - where as​
sinh(x) is monotonic function, with unrestricted range (and domain).​
 

Dr.Peterson

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Nov 12, 2017
Messages
10,590
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
If I were you, I'd just try out both ways and find for myself how they differ. Can you show your work both ways? Are there points in either where you have trouble? Partly it will be a matter of taste or experience.

I'm not sure whether you are saying you didn't follow all the work, or not.
 
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