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

Question

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

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

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

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 $\int (\frac{\sqrt{a^{2} + x^{2}}}{cos (f (x))}) d x$ $int(sqrt{a^2 + x^2}/cos(f(x)))dx$ (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 $d u$ $du$ and $d x$ $dx$ and it all became very messy.

Any advice?

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Answer 1 · 2018-04-10T20:42:57

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 $x^1.18 > 0$

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

1 Answer

Explanation:

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My friend gave me a nasty integral: int sqrt{1 + (-sin(0.7x^{1.18})0.826x^{0.18})} dx∫√1+(−sin(0.7x1.18)0.826x0.18)dxint sqrt{1 + (-sin(0.7x^{1.18})0.826x^{0.18})} dx. Can anybody give me some advice on how to solve this?

1 Answer

Explanation:

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