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



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ganeshie8
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even/odd stuff ha ?

BSwan
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2014 is order of derivative ?

hartnn
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\(\Large \int_a^b f(x) dx= \int_a^bf(a+bx)dx \)
use this!

klimenkov
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@BSwan
2014 is the power.

hartnn
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after you have proved that your function is even

BSwan
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aha cool :)

hartnn
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\(\Large \int_{a}^a f(x) dx = 2\int_0^a f(x)dx\)
if f(x) is even function

hartnn
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did u get what to do ?

klimenkov
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\[2\int\limits_0^{\frac{\pi}2} \frac{\sin^{2014}x}{\sin^{2014}x + \cos^{2014}x} \, dx\]What's next?

hartnn
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\(\Large \int_a^b f(x) dx= \int_a^bf(a+bx)dx\)
replace x by pi/2  x

hartnn
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\(I = 2\int\limits_0^{\frac{\pi}2} \frac{\sin^{2014}x}{\sin^{2014}x + \cos^{2014}x} \, dx ... ... (A)\)
just giving a label, to be used later

klimenkov
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\[2 \int\limits_0^{\frac{\pi}2} \frac{\cos^{2014}x}{\sin^{2014}x + \cos^{2014}x} \, dx\]

klimenkov
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Oops...

hartnn
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\(I = 2 \int\limits_0^{\frac{\pi}2} \frac{\cos^{2014}x}{\sin^{2014}x + \cos^{2014}x} \, dx ... ... (B)\)
Add (A) and (B)

klimenkov
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\[I = 2 \int\limits_0^{\frac{\pi}2} \frac{\cos^{2014}x}{\sin^{2014}x + \cos^{2014}x} \, dx\]

hartnn
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what ? why negative ?

hartnn
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dx remains as dx

hartnn
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i am NOT doing any substitution

klimenkov
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Yeah, everything is okay. My fault.
\[I + I = 2\int_0^{\frac\pi2}dx,\]\[I=\frac\pi2.\]Very nice, thank you.

hartnn
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welcome ^_^