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Evaluate the following: int_(0)^(pi)(1)/(3+2 sin x+cosx)dx |
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Answer» Solution :LET `I=int_(0)^(pi)(1)/(3+2 sin x+cos x)dx` Put `TAN.(x)/(2)=t THEREFORE x =2 tan^(-1)t` `therefore dx=(2dt)/(1+t^(2))and sin x=(2t)/(1+t^(2)), cos x=(1-t^(2))/(1+t^(2))` When x = 0 , t = TNA 0 = 0 When `x=pi, t=tan.(pi)/(2)=oo` `therefore""I=int_(0)^(oo)(1)/(3+2((2)/(1+t^(2)))+((1-t^(2))/(1+t^(2)))).(2dt)/(1+t^(2))` `=int_(0)^(oo)(1+t^(2))/(3+3t^(2)+4t+1-t^(2)).(2dt)/(1+t^(2))` `=2int_(0)^(oo)(1)/(2t^(2)+4t+4)dt` `=(2)/(2)int_(0)^(oo)(1)/(t^(2)+2t+2)dt=int_(0)^(oo)(1)/((t^(2)+2t+1)+1)dt` `=int_(0)^(oo)(1)/((t+1)^(2)+(1)^(2))dt=(1)/(1)[tan^(-1)((t+1)/(1))]_(0)^(oo)` `=[tan^(-1)(t+1)]_(0)^(oo)` `=tan^(-1)oo-tan^(-1)1` `=(pi)/(2)-(pi)/(4)=(pi)/(4).` |
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