`int(x+sqrt(x+1))/(x+2)dx` का मान ज्ञात कीजिए।

Explanation

माना `I =int(x+sqrt(x+1))/(x+2)dx`
यदि `sqrt(x +1) = t, x + 1 = t^(2)`व` dx = 2tdt`
अब `I = int(x+sqrt(x+1))/(x+2)dx = 2int((t^(2) -1 +t)t)/((t^(2) +1))dt`
`=2int((t^(3) + t^(2) - t)/(t^(2) +1))dt = 2int(t+1-(2t+1)/(t^(2) +1))dt`
` = 2int(t+1-(2t)/(t^(2)+1) - (1)/(t^(2) +1))dt`
` = 2int t dt +2 int ddt - 2 int (2t)/(t^(2) +1)dt - 2 int (1)/(t^(2)+1)dt`
`=t^(2) + 2t - 2 log(t^(2) +1) - 2tan^(-1) t+C`
`=(x+1)+2sqrt(x+1)-2log (x+2) - 2tan^(-1) sqrt(x+1)+c`