Mastering Integrals: Tips and Hints for Solving Challenging Equations

[xAxis]

Can anybody give me a hint how to solve, if it is possible at all?
$$\int \frac{dv}{-g-kv\sqrt{v^{2}+u^{2}}}$$

 

[dextercioby]

I don't think it can be solved in terms of elementary functions.

Daniel.

 

[benorin]

I don't think it can be solved in terms of elementary functions.

Daniel.

I'll say, and I quote some computer when I say,

Mathematica could not find a formula for your integral. Most likely this means that no formula exists.

so good job xAxis, you've done quite , no rather , wait! certianly this is . Yes, that's it, I'm quite with this integral/error message combo.

--Ben

 

[dextercioby]

It can be solved exactly in terms of elementary functions.

Daniel.

 

[dextercioby]

$$I=\int \frac{dx}{-g-kx\sqrt{x^{2}+p^{2}}} =-\frac{1}{k}\int \frac{dx}{\frac{g}{k}+x\sqrt{x^{2}+p^{2}}}$$

Now make the substitution

$$x=p\sinh t$$

$$I= -\frac{p}{k}\int \frac{\cosh t \ dt}{\frac{g}{k}+p^{2}\sinh t\cosh t}= -\frac{p}{k}\int \frac{\cosh t \ dt}{\frac{g}{k}+\frac{p^{2}}{2}\sinh 2t}$$,

 

[dextercioby]

$$I= -\frac{p}{k}\int \frac{e^{t} +e^{-t}}{\frac{2g}{k}+\frac{p^{2}}{2}\left(e^{2t}-e^{-2t}\right)} \ dt = -\frac{2}{kp}\left(\int \frac{e^t}{\frac{4g}{p^{2}k}+e^{2t}-e^{-2t}} \ dt + \int \frac{e^{-t}}{\frac{4g}{p^{2}k}+e^{2t}-e^{-2t}} \ dt \right)$$,

$$I=-\frac{2}{kp}\left(\int \frac{e^{3t}}{e^{4t}+\frac{4g}{p^{2}k}e^{2t}-1} \ dt +\int \frac{e^{t}}{e^{4t}+\frac{4g}{p^{2}k}e^{2t}-1} \ dt \right)$$

The 2 remaining integrals can be computed exactly.

Daniel.

 

[dextercioby]

For example:

See the attached file.

Daniel.