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Calculate $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}{\frac{\sin^{2014}x}{\sin^{2014}x + \cos^{2014}x}}dx$$ Integration by parts and variable substitutes don't seem to lead anywhere. Help would be appreciated.

EDIT: a more general version of this question is answered here: How to compute $\int_0^{\pi/2}\frac{\sin^3 t}{\sin^3 t+\cos^3 t}dt$?

Ignacio
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  • Found that a similar question was already solved a long time ago: https://math.stackexchange.com/questions/82489/how-to-compute-int-0-pi-2-frac-sin3-t-sin3-t-cos3-tdt – Ignacio May 22 '18 at 19:43
  • the exponent does not matter, the answer will be the same, hence Victor's remark is justified – imranfat May 22 '18 at 19:58

1 Answers1

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Hints: add and subtract $\cos^{2014}x$ in the numerator. Do a change of variables.

Full work: let $n = 2014$.

$$I=\int_{-\pi/2}^{\pi/2} \frac{\sin^n x + \cos ^n x}{\sin^n x + \cos^n x} dx - \int_{-\pi/2}^{\pi/2} \frac{\cos^n x}{\sin^n x + \cos^n x} dx = \pi - \int_{-\pi/2}^{\pi/2} \frac{\cos^n x}{\sin^n x + \cos^n x} dx $$

Now the rightmost integral is

$$\int_{0}^{\pi} \frac{\cos^n x}{\sin^n x + \cos^n x} dx$$

because the integrand is of period $\pi$. Doing a change of variables $u = x-\pi/2$, we get

$$I = \pi - I, \therefore I = \pi/2$$