How should I integrate $\cos^{2018}(x)/\sin^{2018}(x)+\cos^{2018}(x)$?

Asked Apr 26 '22 at 20:53

Active Apr 26 '22 at 21:03

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I am working on a kind of tricky integral problem that I found online.

$$\int \frac{\cos^{2018}(x)}{\sin^{2018}(x)+\cos^{2018}(x)}dx$$

I tried substitution which changed it to

$$ \cos^{2018}(x) \ \to \sin^{2018}\left(x+\frac\pi2\right) $$

but after this substitution, I am quite uncertain how should I deal with $$\int \frac{\sin^{2018}\left(x+\frac\pi2\right)}{\sin^{2018}(x)+\sin^{2018}\left(x+\frac\pi2\right)}dx$$

I tried every trig identity that I could possibly use in order to sum the integral but I am still struggling with this. How should I approach to solve this integral? It would be very nice if anyone could help with this. Thanks.

edited Apr 26 '22 at 21:03

Samuel Adrian Antz

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asked Apr 26 '22 at 20:53

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Weierstraß substitution might help: https://en.wikipedia.org/wiki/Weierstrass_substitution – Samuel Adrian Antz Apr 26 '22 at 21:03
The indefinite integral gets messy even for much smaller exponents. – Dan Apr 26 '22 at 21:27
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Are you sure the integration limits are not given? – Átila Correia Apr 26 '22 at 22:14
You're integrating $\dfrac{1}{\tan^{2n}+1}$ which has period $\pi$. For large $n$ this is essentially a square wave. – John Wayland Bales Apr 27 '22 at 02:15
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Surely there is no nice explicit form for the indefinite integral. If you're looking for the definite integral $\int_0^{\pi/2}$, that has been asked many times before; see here for example. – Hans Lundmark Apr 27 '22 at 05:59

How should I integrate $\cos^{2018}(x)/\sin^{2018}(x)+\cos^{2018}(x)$?

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