InterviewSolution
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InterviewSolution
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`int(1)/(x^(4)+1)dx` का मान ज्ञात कीजिए । |
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Answer» माना `I=int(1)/(x^(4)+1)dx=(1)/(2)int(2)/(x^(4)+1)dx` `=(1)/(2)int((x^(2)+1)-(x^(2)-1))/(x^(4)+1)dx` `=(1)/(2)int(x^(2)+1)/(x^(4)+1)dx-(1)/(2)int(x^(2)-1)/(x^(4)+1)dx` `=(1)/(2)int(1+(1)/(x^(2)))/(x^(2)+(1)/(x^(2)))dx-(1)/(2)int(1-(1)/(x^(2)))/(x^(2)+(1)/(x^(2)))dx` `rArr" "I=(1)/(2)int(1+(1)/(x^(2)))/((x-(1)/(x))^(2)+(sqrt2)^(2))dx-(1)/(2)int(1-(1)/(x^(2)))/((x+(1)/(x))^(2)-(sqrt2)^(2))dx` माना प्रथम समाकल के लिये `x-(1)/(x)=t rArr (1+(1)/(x^(2)))dx=dt` और दूसरे समाकल के लिये `x+(1)/(x)=u rArr (1-(1)/(x^(2)))dx=du` `therefore I=(1)/(2)int(dt)/(t^(2)+(sqrt2)^(2))-(1)/(2)int(du)/(u^(2)-(sqrt2)^(2))` `=(1)/(2sqrt2)tan^(-1).(t)/(sqrt2)-(1)/(2).(1)/(2sqrt2)log|(u-sqrt2)/(u+sqrt2)|+c` `=(1)/(2sqrt2)tan^(-1)((x-(1)/(x))/(sqrt2))-(1)/(4sqrt2)log|(x+(1)/(x)-sqrt2)/(x+(1)/(x)+sqrt2)|+c` `=(1)/(2sqrt2)tan^(-1)((x^(2)-1)/(xsqrt2))-(1)/(4sqrt2)log|(x^(2)-xsqrt2+1)/(x^(2)+xsqrt2+1)|+c` |
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