`int (x^(2)+1)/(x^(4)+x^(2)+1)dx` का मान ज्ञात कीजिए।

Options

A. `(1)/(sqrt(3))tan^(-1)((x^(2)-1)/(xsqrt(3)))+C`
B. `(1)/(sqrt(3))tan^(-1)((x^(2)-1)/(x))+C`
C. `tan^(-1)((x^(2)-1)/(xsqrt(3)))+C`
D. `tan^(-1)((x^(2)-1)/(x))+C`

Answer

A

Explanation

दिया है, `int (x^(2)+1)/(x^(4)+x^(2)+1)dx= int (1+1//x^(2))/(x^(2)+(1)/(x^(2))+1)dx`
`= int ((1+(1)/(x^(2))))/((x^(2)+(1)/(x^(2))-2))dx=int ((1+(1)/(x^(2))))/((x-(1)/(x))+3)dx`
माना `t=x-(1)/(x)rArr dt=(1+(1)/(x^(2)))dx`
`:. int (dt)/(t^(2)+(sqrt(3)^(2)))=(1)/(sqrt(3))tan^(-1)((t)/(sqrt(3)))+C`
`=(1)/(sqrt(3)) tan^(-1)[(1)/(sqrt(3))*((x^(2)-1))/(x)]+C`