InterviewSolution
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InterviewSolution
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निम्नलिखित को सिद्ध कीजिए । `int_(1)^(3)(dx)/(x^(2)(x+1))=(2)/(3)+log.(2)/(3)` |
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Answer» `int_(1)^(3)(1)/(x^(2)(x+1))dx=(2)/(3)+log.(2)/(3)` माना `" "(1)/(x^(2)(x+1))=(A)/(x)+(B)/(x^(2))+(C)/(x+1)" …(1)"` `rArr" "1=Ax(x+1)+B(x+1)+Cx^(2)" …(2)"` x = 0 तो `1=0+B+0" "rArr" "B=1"` `x = -1` तो `1=0+0+C" "rArr" "C=1` `x^(2)` के गुणांकों की तुलना करने पर , `0=A+C" "rArr" "A=-C=-1` `therefore int_(1)^(3)(1)/(x^(2)(x+1))dx` `=int_(1)^(3)(-(1)/(x)+(1)/(x^(2))+(1)/(x+1))dx` `=[-log|x|-(1)/(x)+log|x+1|]_(1)^(3)` `=[log|(x+1)/(x)|-(1)/(x)]_(1)^(3)` `=(log|(4)/(3)|-(1)/(3))-(log|(2)/(1)|-(1)/(2))` `=(log.(4)/(3)-log2)+(1-(1)/(3))` `=log((4)/(3)xx(1)/(2)+(2)/(3)=log.(2)/(3)+(2)/(3))` `" "`यही सिद्ध करना था । |
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