InterviewSolution
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InterviewSolution
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`int((x^(2)+1)dx)/((x+1)^(3)(x-2))` का मान ज्ञात कीजिए। |
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Answer» माना `" "I=int((x^(2)+1)dx)/((x+1)^(3)(x-2))` माना `" "x+1=t rArr x = t-1" तथा " dx=dt` `therefore" "I=int((t-1)^(2)+1)/(t^(3)(t-3))dt=(1)/(t^(3))int(t^(2)-2t+2)/((t-3))dt` अब `t^(2)-2t+2` को `t-3` से भाग दीजिये तथा t की घातों को आरोही क्रम (ascending order ) में रखने पर `=(1)/(t^(3))[-(2)/(3)+(4)/(9)t-(5)/(27)t^(2)+(5t^(3))/(27(t-3))]` `=[-(2)/(3)t^(-3)+(4)/(9)t^(-2)-(5)/(27).(1)/(t)+(5)/(27(t-3))]` `therefore" "int(t^(2)-2t+2)/(t^(3)(t-3))dt=(-2)/(3)intt^(-3)dt+(4)/(9)int t^(-2)dt-(5)/(27)int(dt)/(t)+(5)/(27)int(dt)/((t-3))` `=-(2)/(3)(t^(-2))/(-2)+(4)/(9)(t^(-1))/(-1)-(5)/(27)log t+(5)/(27)log(t-3)` `=(5)/(27)[log(t-3)-logt]+(1)/(3t^(2))-(4)/(9t)` `=(5)/(27)[log(x-2)-log(x+1)]+(1)/(3(x+1)^(2))-(4)/(9(x+1))` `=(5)/(27)log((x-2)/(x+1))+(1)/(3(x+1)^(2))-(4)/(9(x+1))` |
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