इन समाकलनो का मान ज्ञात कीजिये
`underset(1)overset(2)int (5x^(2))/(x^(2)+4x+3)`

Explanation

माना `I=int_(1)^(2)(5x^(2))/(x^(2)+4x+3)dx`
यहाँ, `" "(5x^(2))/(x^(2)+4x+3)=5-(20x+15)/(x^(2)+4x+3)`
`therefore" "I=int_(1)^(2)(5-(20x+15)/(x^(2)+4x+3))dx`
`=int_(1)^(2)5dx-int_(1)^(2)(20x+15)/((x+1)(x+3))dx" …(1)"`
माना `" "(20x+15)/((x+1)(x+3))dx=(A)/((x+1))+(B)/((x+3))`
`rArr" "20x+15=A(x+3)+B(x+1)`
`x=-1" तो "-20+15=A(-1+3)+0`
`rArr" "A=-(5)/(2)`
`x=-3" तो "-60+15=0+B(-3+1)`
`rArr" "B=(45)/(2)`
`therefore" "I=5int_(1)^(2)1dx+(5)/(2)int_(1)^(2)(1)/((x+1))dx-(45)/(2)int_(1)^(2)(1)/((x+3))dx`
`" "` [समीकरण (1 ) से ]
`=[5x+(5)/(2)log|x+1|-(45)/(2)log|x+3|]_(1)^(2)`
`=(10+(5)/(2)log|3|-(45)/(2)log|5|-5-(5)/(2)log|2|+(45)/(2)log|4)`
`rArr" "I=5+(5)/(2)log|(3)/(2)|-(45)/(2)log|(5)/(4)|`