ज्ञात करें `int ( 2x)/((1+x^2)(3+x^2))dx`

Explanation

माना की `I =int ( 2x)/((1+x^2)(3+x^2))dx`
`x^2 =t` रखे, तो `2x dx = dt`
`:." "I = int (dt)/((1+t)(3+t))=1/2 int [1/(1+t)-1/(3+t)]dt" "` [ partial freaction से ]
` =1/2 [ log | 1+t | -log | 3+t|] +c =1/2 log |(1+t)/(3+t)|+c`
`=1/2log|(1+x^2)/(3+x^2)|+c=1/2 log((1+x^2)/(3+x^2))+c (because1+x^2 gt 0, 3+x^3 gt 0)`