\[\int_{-\frac{\pi}2}^{\frac{\pi}2} \frac{\sin^{2014}x}{\sin^{2014}x + \cos^{2014}x} \, dx\]

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\[\int_{-\frac{\pi}2}^{\frac{\pi}2} \frac{\sin^{2014}x}{\sin^{2014}x + \cos^{2014}x} \, dx\]

Mathematics
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even/odd stuff ha ?
2014 is order of derivative ?
\(\Large \int_a^b f(x) dx= \int_a^bf(a+b-x)dx \) use this!

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Other answers:

@BSwan 2014 is the power.
after you have proved that your function is even
aha cool :)
\(\Large \int_{-a}^a f(x) dx = 2\int_0^a f(x)dx\) if f(x) is even function
did u get what to do ?
\[2\int\limits_0^{\frac{\pi}2} \frac{\sin^{2014}x}{\sin^{2014}x + \cos^{2014}x} \, dx\]What's next?
\(\Large \int_a^b f(x) dx= \int_a^bf(a+b-x)dx\) replace x by pi/2 - x
\(I = 2\int\limits_0^{\frac{\pi}2} \frac{\sin^{2014}x}{\sin^{2014}x + \cos^{2014}x} \, dx ... ... (A)\) just giving a label, to be used later
\[2 \int\limits_0^{\frac{\pi}2} \frac{\cos^{2014}x}{\sin^{2014}x + \cos^{2014}x} \, dx\]
Oops...
\(I = 2 \int\limits_0^{\frac{\pi}2} \frac{\cos^{2014}x}{\sin^{2014}x + \cos^{2014}x} \, dx ... ... (B)\) Add (A) and (B)
\[I = -2 \int\limits_0^{\frac{\pi}2} \frac{\cos^{2014}x}{\sin^{2014}x + \cos^{2014}x} \, dx\]
what ? why negative ?
dx remains as dx
i am NOT doing any substitution
Yeah, everything is okay. My fault. \[I + I = 2\int_0^{\frac\pi2}dx,\]\[I=\frac\pi2.\]Very nice, thank you.
welcome ^_^

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