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?

1 Answer
Apr 10, 2018

Please see below.

Explanation:

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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:

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For gamma > 0

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

enter image source here