I was looking at some integration problems the other day and I came across this identity:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\int_{0}^{\frac{\pi}{2}} \sin^{p}x \cos^{q}x dx = \frac{1}{2} \mbox{B} \left( \frac{p+1}{2},\frac{q+1}{2}\right)[/tex]

where B(x,y) is the Beta function, for Re(x) and Re(y) > 0. From the way in which the above formula is derived, it turns out that this is valid for p > -1 and q > -1.

However, I'm interested in seeing if this identity can be extended to allow any real power of p or q. Does anyone know of a similar identity that enables me to solve this for any real p or q, or over any interval?

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# Beta Function Extension?

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