InterviewSolution
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InterviewSolution
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`int(1)/(1+x^(3))` का मान ज्ञात कीजिए। |
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Answer» माना `(1)/(1+x^(3))=(1)/((x+1)(x^(2)-x+1))` `" "=(A)/(x+1)+(Bx+C)/(x^(2)-x+1)` `" "=(A(x^(2)-x+1)+(Bx+C)(x+1))/((x+1)(x^(2)-x+1))` `rArr" "A(x^(2)-x+1)+(Bx+C)(x+1)=1` समान घातों के गुणांकों की तुलना करने पर `A+B=0` `-A+B+C=0` `A+C=1` हल करने पर `A=(1)/(3),B=-(1)/(3),C=(2)/(3)` `therefore int(1)/(1+x^(3))dx` `=(1)/(3)int(1)/(1+x^(3))dx` `=(1)/(3)int(1)/(x+1)dx+int(-(1)/(3)x+(2)/(3))/(x^(2)-x+1)dx` `=(1)/(3)log|x+1|-(1)/(3)int(x-2)/(x^(2)-x+1)` `=(1)/(3)log|x+1|-(1)/(6)int(2x-4)/(x^(2)-x+1)dx` `=(1)/(3)log|x+1|-(1)/(6)int((2x-1)-3)/(x(2)-x+1)dx` `=(1)/(3)log|x+1|-(1)/(6)int(2x-1)/(x^(2)-x+1)dx+(1)/(2)int(1)/(x^(2)-x+1)dx` `=(1)/(3)log|x+1|-(1)/(6)log|x^(2)-x+1|+(1)/(2)int(1)/((x-(1)/(2))^(2)+((sqrt3)/(2))^(2))dx` `=(1)/(3)log|x+1|-(1)/(6)log|x^(2)-x+1|+(1)/(2).(1)/(sqrt3//2)tan^(-1).(x-(1)/(2))/((sqrt3)/(2))+c` `=(1)/(3)log|x+1|-(1)/(6)log|x^(2)-x+1|+(1)/(sqrt3)tan^(-1).(2x-1)/(sqrt3)+c` |
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