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Small note on recursive formula for integral of trigonometric functions

Nasser M. Abbasi

June 29, 2015 page compiled on June 29, 2015 at 11:43pm

After stuggling in deriving this, I found similar one on wikpedia. References below. May be I will add Mathematica implementation for this later....

The goal is to find recusive formula for ∫       n
  cos(x) dx  . Starting by rewriting it as

∫               ∫
   cos(x)ndx =    cos(x)n−1 cos(x)dx
(1)

Integrating by parts ∫               ∫
  udv =  (uv) −   vdu  and letting           n− 1
u = cos(x )   ,dv = cos(x)  , hence du =  − (n − 1 )cos(x)n−2sin(x)  and v = sin (x)  the above becomes

pict

The ∫
 cos(x )ndx  in the RHS above is what is being solved for. Moving it to the LHS gives

pict

Therefore the recusrive formula is

∫                     n−1                 ∫
   cos(x)ndx =  cos(x)---sin(x)-+ (n-−-1)    cos(x )n−2dx
                       n             n

References:

1.
http://www.integraltec.com/math/math.php?f=cosPower.html#cos
2.
http://en.wikipedia.org/wiki/Integration_by_reduction_formulae#Examples