`int(2x+1)/(sqrt(2x^(2)+x-3))dx` का मान ज्ञात कीजिए ।

Explanation

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 ")`