`(6x+7)/(sqrt((x-5)(x-4)))`

Explanation

माना `I=int(6x+7)/(sqrt((x-5)(x-4)))dx`
`=int(6x+7)/(sqrt(x^(2)-9x+20))dx`
माना `6x+7=A(d)/(dx)(x^(2)-9x+20)+B`
`rArr" "6x+7=A(2x-9)+B`
x के गुणकों की तुलना करने पर ,
`6=2A" "rArr" "A=3`
अचर पदों की तुलना करने पर ,
`" "7=-9A+B`
`rArr" "7=-27+B`
`rArr B=34`
`therefore" "I=int(3(2x-9)+34)/(sqrt(x^(2)-9x+20))" "` प्रथम समाकलन के लिये
`=3int(2x-9)/(sqrt(x^(2)-9x+20))dx+34int(1)/(sqrt(x^(2)-9x+20))dx" "` माना `x^(2)-9x+20=t`
`" "rArr " "(2x-9)dx=dt`
`=3int(1)/(sqrtt)dt+34int(1)/(sqrt((x^(2)-9x+(81)/(4))+20-(81)/(4)))dx`
`=3intt^(-1//2)dt+34int(1)/(sqrt((x-(9)/(2))^(2)-((1)/(2))^(2)))dx`
`=(3.t^(1//2))/(1//2)+34log|x-(9)/(2)+sqrt((x-(9)/(2))^(2)-((1)/(2))^(2))|+c`
`=6sqrtt+34log|x-(9)/(2)+sqrt(x^(2)-9x+20)|+c`
`=6sqrt(x^(2)-9x+20)+34log|x-(9)/(2)+sqrt(x^(2)-9x+20)|+c`