reduction formulas for integration of powers

The following reduction formulas, with integer n and via integration by parts, may be used for lowing (n>0) or raising (n<0) the the powers:

  • sinnxdx=-1nsinn-1xcosx+n-1nsinn-2xdx  (n0)

  • cosnxdx=1ncosn-1xsinx+n-1ncosn-2xdx  (n0)

  • (lnx)ndx=x(lnx)n-n(lnx)n-1dx  (n0)

  • 1(1+x2)ndx=12n-2x(1+x2)n-1+2n-32n-21(1+x2)n-1dx(n>1)

Example.  For finding dxsin3x, we apply the first formula with  n:=-1,  getting first


From this we solve


(see integration of rational function of sine and cosine).

Note 1.  Instead of the two first formulae, it is simpler in the cases when n is a positive odd or a negative even numberMathworldPlanetmath to use the following
which may be found after making the powers on the right hand sides to polynomialsPlanetmathPlanetmath.

Note 2.tannxdx  (n+)  is obtained easily by the substitution (  tanx:=t,  dx=dtt2+1  and a division; e.g.

tan5xdx =t5t2+1𝑑t=(t3-t+tt2+1)𝑑t
Title reduction formulas for integration of powers
Canonical name ReductionFormulasForIntegrationOfPowers
Date of creation 2013-03-22 18:36:53
Last modified on 2013-03-22 18:36:53
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Topic
Classification msc 26A36
Classification msc 26A09
Synonym integration of powers
Related topic GeneralFormulasForIntegration
Related topic IntegralTables
Related topic WallisFormulae