Find the value of the following Integral
$$2^{2010} \frac{\int_0^{1} x^{1004}(1-x)^{1004} dx}{\int_0^{1} x^{1004}(1-x^{2010})^{1004} dx}$$
Because it is in power of 1004 i am not able to find the correct substitution.
Value of $2^{2010} \frac{\int_0^1 x^{1004}(1-x)^{1004} dx}{\int_0^1 x^{1004}(1-x^{2010})^{1004} dx}$
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For the denominator, $x^{1005}=u$ and for the numerator $x=\sin t$ – lab bhattacharjee Oct 28 '17 at 03:25
1 Answers
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Hint
For the numerator, consider $$I_n=\int x^n(1-x)^n\,dx$$ Let $x=\sin^2(t)$, $dx=2\sin(t)\cos(t)$ to make $$I_n=\frac 1 {2^{2n}}\int \sin^{2n+1}(2t)\,dt$$ Have a look here.
Claude Leibovici
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This can be solved more easily by the application of the beta function. – user 493905 Dec 15 '19 at 09:46
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I agree with you for sure. I did not know if the OP already knew it. Cheers :-) – Claude Leibovici Dec 15 '19 at 09:50