# A tip for integral please

1. Nov 20, 2006

### xAxis

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

Last edited: Nov 20, 2006
2. Nov 21, 2006

### dextercioby

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

Daniel.

3. Nov 22, 2006

### benorin

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

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

--Ben

4. Nov 22, 2006

### dextercioby

It can be solved exactly in terms of elementary functions.

Daniel.

5. Nov 22, 2006

### 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

$$x=p\sinh t$$

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

6. Nov 22, 2006

### 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.

Last edited: Nov 22, 2006
7. Nov 22, 2006

### dextercioby

For example:

See the attached file.

Daniel.

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