`int (x+2)/(2x^(2)+6x+5)dx` का मान ज्ञात कीजिए।

Options

A. `(1)/(4)log |2x^(2)+6x+5|+(1)/(2)tan^(-1)(2x+3)+C`
B. `(1)/(4)log|2x+3|-(1)/(2)tan^(-1)(2x^(2)+6x+5)+C`
C. `(1)/(2)log|2x^(2)+6x+5|-(1)/(4)tan^(-1)(2x+3)+C`
D. `(-1)/(2)log|2x^(2)+6x+5|-(1)/(5)tan^(-1)(2x+3)+C`

Answer

A

Explanation

माना `I= int (x+2)/(2x^(2)+6x+5)dx int (A(d)/(dx)(2x^(2)+6x+5)+8)/(2x^(2)+6x+5)`
`rArr x+2=A(d)/(dx)(2x^(2)+6x+5)+B rArr x+2=A(4x+6)B`
`:. A=1//4` तथा `B=1//4`
अतः `I= int ((1)/(4)(4x+6)+(1)/(2))/(2x^(2)+6x+5)dx= int ((1)/(4)(4x+6))/(2x^(2)+6x+5)dx+(1)/(2x^(2)+6x+5)dx`
`=(1)/(4)log|2x^(2)+6x+5|+(1)/(4) int (1)/(x^(2)+3x+(5)/(2))dx" ....(ii)`
`I_(1)=(1)/(4)int (1)/(x^(2)+3x+(5)/(2))dx`
`=(1)/(4)int (1)/((x+(3)/(2))^(2)-(9)/(4)+(5)/(2))dx=(1)/(4)int (1)/((x+(3)/(2))^(2)+((1)/(2))^(2))dx`
`((1)/(4))/((1)/(2)) "tan"^(-1)((x+(3)/(2)))/((1)/(2))+C =(1)/(2) tan^(-1)(2x+3)+C`
समी (i) से `I=(1)/(2)log|2x^(2)6x+5|+(1)/(2)tan^(-1) (2x+3)+C`