# Help solving complicated integral

1. Jan 10, 2010

### elkface

This integral is not overly complicated, but I am having a little trouble with it - I haven't been in calc for a year almost, and it seems I have lost my integration abilities with it.

Here's the integral:
$$\int \sqrt{1 + (a \pi x^{a-1})^{2} \cos^{2}({\pi x^{a}})} dx$$

I figure the best place to start is to factor out the inside expression and get rid of the square root, but even doing this I am lost.

Any help?

2. Jan 12, 2010

### CFDFEAGURU

Can you show your steps so far on your attempt to solve this integral?

Thanks
Matt

3. Jan 14, 2010

### elkface

Well, to be honest, all I can seem to do is stare at it. I can't factor it, clearly, to get it separated into parts so that I could even attempt to do int. by parts or anything of that sort. I have tried Wolfram online integrator, but that doesn't work because it takes the server too long to process.

If anyone has software that could integrate this, or can recommend some (preferably freeware), I'd appreciate it. I don't need the work - I just need the answer.

4. Jan 15, 2010

### HallsofIvy

If you replace $cos^2$ by $1- sin^2$ that becomes an elliptic integral which cannot be integrated in terms of elementary functions.

5. Jan 15, 2010

### elkface

Ok, thanks for the tip. In that case, I will probably not be able to put this integral to any good use, as I am not familiar with most non-elementary functions. Thanks anyway.

6. Jan 15, 2010

### CFDFEAGURU

Here is a link to the Numerical Recipe C routine for solving eliptical integrals. Not sure if if will help you. Take caution when using the NR routines they can be problematic. You have been warned.

http://www.nrbook.com/a/bookcpdf/c6-11.pdf

Thanks
Matt