power reduction formula for the integral of sin^n(x)

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[kshitizmangalbajracharya]

There's something called "Reduction Formulae" in (really) old European textbooks on Integral Calculus. It's still there in undergraduate Calculus course in Nepal and probably in many parts of India. The topic basically deals with such integrals and many other similar integrals where we need to reduce the power. It can also involve two powers out of which only one is reduced. Two or more powers are also possible, obviously.

[holyshit]

... and rewrite it into the sum
Without reduction
Substitute u=Pi/2-x to get cosine
Calculate coefficients of Chebyshov polynomial of degree n
Put nonzero coefficients of Chebyshov polynomials of degree k where k is natural number not grater then n which gives the same rest in division by 2 as n gives
columnwise in the matrix
Inverse that matrix
Now you can integrate, but integration is very easy