From: rusin@washington.math.niu.edu (Dave Rusin) Newsgroups: sci.math Subject: Re: integral sin(x)^n dx Date: 1 Sep 1995 18:59:46 GMT In article , Gammel, ing. A.M. GST wrote: >Can anyone supply a symbolic solution or a progression for the following >Integral (for arbitrary n) > > Integral [ (sin(x)+1)^n ]dx Let u=pi/2 - x; this is then -Integral [ (cos(u)+1)^n ] du But cos(u)+1 = 2 cos^2(u/2), so setting v = u/2 we get - 2^(n+1) Integral [ cos^(2n) (v) ] dv Probably the easiest way to remember integrals for powers of cosine (or sine) is to use cos(v) = (1/2) ( exp(iv) + exp(-iv) ); thus cos^(2n) (v) = (1/2)^(2n) ( Sum[ (2n choose j) exp( i(2n-2j)v ), j = 0..2n ] ), which integrates to (the case j=n is separate): (1/2)^(2n) ( Sum[ (2n choose j) exp( i(2n-2j)v )/(2i)(n-j), j = 0..2n, j<>n ] + (2n choose n)v ), Collecting pairs of terms gives (1/2)^(2n) ( Sum[ (2n choose j) sin( (2n-2j)v )/(n-j), j=0..n-1 + (2n choose n) v ) So I get the original integral to be (up to an additive constant) 1/2^n ( (2n choose n) x + Sum[ -2(2n choose j)/(n-j) sin((n-j)(pi/2-x)), j = 0..n-1 ] ) Of course you can rewrite the trig functions to remove the pi's, or you can express everything as a polynomial in sin(x) and cos(x). dave ============================================================================== From: dray@leland.stanford.edu (Edray Herber Goins) Newsgroups: sci.math Subject: Re: integral sin(x)^n dx Date: Fri, 01 Sep 1995 14:35:07 -0700 You can use the relation 1 + sin(x) = 2 (sin[x/2 + Pi/4]^2) to express the integral in a simpler form. Define I[n] = Integral[ (1 + sin(x))^n dx] Assuming that we will integrate from -(Pi/2) to x, we can write the integral in the form I[n] = 2^(n+1) Integral[ sin(X)^(2n) dX] where we integrate from 0 to X, where X = (x/2) + (Pi/4). It is easy to show (by integrating by parts, for example) that for n = 1, 2, 3, ... n I[n] = (2n-1) I[n-1] - 2^n sin(X)^(2n-1) cos(X) where I[0] = x + Pi/2 and X is given as before. From here, you should be able to find a closed form solution for I[n] by induction. -- Stanford University Department of Mathematics - Building 380 Stanford, CA 94305-2125