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what is integration of 1/(a^2-2*a*cos(x)+1) its lower limit is 0 and upper limit is pie
I= dx/[a2 -2acosx +1] lim 0 to pi put cosx = 1-tan2x/2/1+tan2x/2 I= sec2x/2 dx / [(a-1)2 + (a+1)2tan2x/2] now put tanx/2 = t sec2x/2dx =2dt I = 2dt / [(a-1)2+(a+1)2t2) I =2dt/(1+a)2 .[(a-1/a+1)2 + t2] this integral is same as 1/a2+x2 dx & its integral is tan-1(x/a)/a so I =2/(a2-1)tan-1[(a+1)t/(a-1)] I = 2/(a2-1)tan-1[(a+1)tanx/2/(a-1)] lim 0 tp pi taking limit I =2/(a2-1) [ tan-1infinity -tan-10] =pi/(a2-1) ans
I= dx/[a2 -2acosx +1] lim 0 to pi
put cosx = 1-tan2x/2/1+tan2x/2
I= sec2x/2 dx / [(a-1)2 + (a+1)2tan2x/2]
now put tanx/2 = t
sec2x/2dx =2dt
I = 2dt / [(a-1)2+(a+1)2t2)
I =2dt/(1+a)2 .[(a-1/a+1)2 + t2]
this integral is same as 1/a2+x2 dx & its integral is tan-1(x/a)/a
so
I =2/(a2-1)tan-1[(a+1)t/(a-1)]
I = 2/(a2-1)tan-1[(a+1)tanx/2/(a-1)] lim 0 tp pi
taking limit
I =2/(a2-1) [ tan-1infinity -tan-10]
=pi/(a2-1) ans
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