Solve the following integrals:
$$ \int_0^{\frac{ \pi }{2}} \frac{\sin^{1395}x}{\sin^{1395}x + \cos^{1395}x}\ dx $$
And it does not follow a specific pattern.
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3I have to ask, where did this question come from? – Zachary F May 06 '16 at 17:23
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2The frequency of questions based on this trick seems to be increasing... – Did May 06 '16 at 17:53
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Recall this useful formula: $$\int_a^b f(x)\ \mathrm dx=\int_a^b f(a+b-x)\ \mathrm dx$$
Let the required value of the integral be $I$. Now, we have: $$ I=\int_0^{\frac{ \pi }{2}} \frac{\sin^{1395}x}{\sin^{1395}x + \cos^{1395}x}\ \mathrm dx = \int_0^{\frac{ \pi }{2}} \frac{\sin^{1395}(\frac\pi2-x)}{\sin^{1395}(\frac\pi2-x) + \cos^{1395}(\frac\pi2-x)}\ \mathrm dx$$
Simplifying: $$I=\int_0^{\frac{ \pi }{2}} \frac{\cos^{1395}x}{\sin^{1395}x + \cos^{1395}x}\ \mathrm dx$$
Adding the two equations: $$2I=\int_0^{\frac{ \pi }{2}} \frac{\sin^{1395}x+\cos^{1395}x}{\sin^{1395}x + \cos^{1395}x}\ \mathrm dx=\int_0^{\frac\pi2}\mathrm dx=\frac\pi2$$
Therefore: $$I=\frac\pi4$$
Kenny Lau
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