InterviewSolution
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InterviewSolution
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`int_(0)^(pi//2)(dx)/((4sin^(2)x+5cos^(2)x))` का मान ज्ञात कीजिए । |
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Answer» `int_(0)^(pi//2)(dx)/((4sin^(2)x+5cos^(2)x))` अंश व हर को `cos^(2)x` से भाग देने पर, `int_(0)^(pi//2)(dx)/((4sin^(2)x+5cos^(2)x))=int_(0)^(pi//2)(sec^(2)x)/((4tan^(2)x+5))dx` यदि `tanx=t rArr sec^(2) xdx = dt` व `x=0 rArr t = 0` व `x=(pi)/(2)rArr t=oo,` तब `=int_(0)^(oo)(dt)/((4t^(2)+5))` `=(1)/(4)int_(0)^(oo)(dt)/(t^(2)+((sqrt5)/(2))^(2))=(1)/(4).(2)/(sqrt5)[tan^(-1).(2t)/(sqrt5)]_(0)^(oo)` `=(1)/(2sqrt5[tan^(-1)(oo)-tan^(-1)(0)])` `=(1)/(2sqrt5)((pi)/(2)-0)=(pi)/(4sqrt5)` |
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