ज्ञात करें `int (dx)/(x^3 +x^2+x+1)`

Explanation

माना की `I = int (dx)/(x^3 +x^2+x+1)`
यहां `x^3 +x^2 +x+1=x^2(x+1)+(x+1)=(x+1)(x^2+1)`
`:." "1/(x^3 +x^2 +x+1) = 1/(x+1)(x^2+1)-= A/(x+1)+(Bx+C)/(x^2+1)`
`rArr" "1-= A(x^2 +1) +(Bx+C)(x-1)`
`x=0, 1, -1` रखने पर मिलता है
`1=A+C, 1=2A +2B +2C` तथा `1=2A`
`:. "A=1/2, B=-1/2, C=1/2" ":." "1/(x^3 +x^2+x+1)=1/2 ""1/((x+1))+(1/2(-x+1))/(x^2+1)`
`:." "I = int (dx)/(x^3 + x^2 +x+1) =1/2 int (dx)/(x+1)-1/2 int x/(x^2+1) dx +1/2 int (dx)/(x^2+1)+c`
`=1/2 log|x+1| -1/4 log |x^2+1|+1/2tan^(-1) x+c`