ज्ञात करें `int sqrt(2x^2 + 3x +4) dx`

Explanation

`I = int sqrt(2x^2 +3x +4)dx = intsqrt(2(x^2+3/2x+2))dx`
` = sqrt2 int sqrt(x^2 +3/2x+2)dx`
`=sqrt2 int sqrt(x^2+2*x""3/4+(3/4)^2-(3/4)^2+2)dx`
`=sqrt2 int sqrt((x+3/4)^2 +23/16)dx" ...(1)"`
माना की `z = x +3/4,` तो `dz = dx`,माना कि `sqrt(23/16)=sqrt(23)/4=a`
तो (1) से, `I = sqrt2 int sqrt( z^2 +a^2)`dz
अब, `int sqrt(z^2+a^2) dz = z/2 sqrt(z^2+a^2) + a^2/2 log (z +sqrt(z^2 +a^2))+c`
`=((x+3/4))/2sqrt((x+3/4)^2+23/16+23/16) +23/32 log [x+3/4+sqrt((x+3/4)^2+23/16)]+c`
`:." "` (1) से, `I =(4x+3)/3sqrt(2x^2+3x+4)+23/(16sqrt2)log [(4x+3)/4+sqrt(2x^2+3x+4)/sqrt2]+c`