Question

My friend gave me a nasty integral: `int sqrt{1 + (-sin(0.7x^{1.18})0.826x^{0.18})} dx`. Can anybody give me some advice on how to solve this?

`int sqrt{1 + (-sin(0.7x^{1.18})0.826x^{0.18})} dx`

I'm having a bit of trouble with this one. I've tried fitting this into the form `int sqrt{a^2 + x^2} dx`.

However, it's a bit tricky considering the fact that there is an inner trigonometric function that has to be dealt with. I tried substitution leaving me with something with the form `int(sqrt{a^2 + x^2}/cos(f(x)))dx` (where f is some function of x) and then I tried integration by parts but I got lost since I had to mix integrals with `du` and `dx` and it all became very messy.

Any advice?

Answer 1

Please see below.

Explanation:

.

This is a difficult integral that will result in an incomplete `Gamma` Function. You can use Wolframalpha.com to obtain the result. Two different forms of the answer are shown below:

For `gamma > 0`

For `0 < gamma < 1` and `x^1.18 > 0`