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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

what is integration of 1/(a^2-2*a*cos(x)+1) its lower limit is 0 and upper limit is pie

Grade:12

1 Answers

vikas askiitian expert
509 Points
10 years ago

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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