InterviewSolution
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InterviewSolution
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`int(2x+1)/(sqrt(2x^(2)+x-3))dx` का मान ज्ञात कीजिए । |
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Answer» माना `I=int(2x+1)/(sqrt(2x^(2)+x-3))dx` माना `2x+1=A.(d)/(dx)(2x^(2)+x-3)+B` `=A(4x+1)+B` x के गुणांकों की और अचर पदों की तुलना करने पर `{:(,2=4A," और ",1=A+B),(rArr.,A=1//2," और ",B=1//2):}` अब `I=(1)/(2)int(4x+1)/(sqrt(2x^(2)+x-3))dx+(1)/(2)int(1)/(sqrt(2x^(2)+x-3))dx` `=I_(1)+I_(2)` (माना ) `{:(I_(1),(1)/(2)int(4x+1)/(sqrt(2x^(2)+x-3))dx," माना ",2x^(2)+x-3=t),(,=(1)/(2)int(1)/(sqrtt).dt," "rArr.,4x+1=(dt)/(dx)),(,=sqrtyt," "rArr.,(4x+1)dx=dt),(,=sqrt(2x^(2)+x-3),,):}` और `I_(2)=(1)/(2)int(1)/(sqrt(2x^(2)+x-3))dx` `=(1)/(2sqrt2)int(1)/(sqrt(x^(2)+(1)/(2)x-(3)/(2)))dx` `=(1)/(2sqrt2)int(1)/(sqrt((x^(2)+(1)/(2)x+(1)/(16))-((3)/(2)+(1)/(16))))dx` `=(1)/(2sqrt2)int(1)/(sqrt((x+(1)/(4))^(2)-((5)/(4))^(2)))dx` `=(1)/(2sqrt2)log|x+(1)/(4)+sqrt((x+(1)/(4))^(2)-((5)/(4))^(2))|` `=(1)/(2sqrt2)log|x+(1)/(4)+sqrt(x^(2)+(1)/(2)x-(3)/(2))|` `=(1)/(2sqrt2)log|(4x+1+sqrt(16x^(2)+8x-24))/(4)|` `=(1)/(2sqrt2)log|4x+1+sqrt(16x^(2)+8x-24)|-(1)/(2sqrt2)log4` `therefore I=sqrt(2x^(2)+x-3)+(1)/(2sqrt2)log|4x+1+sqrt(16x^(2)+8x-24)|+c` `(because -(1)/(2sqrt2)log4 "अचर है ")` |
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