`int(x^(4)+1)/(x^(6)+1)dx` का मान ज्ञात कीजिए ।

Explanation

माना `I=int(x^(4)+1)/(x^(6)+1)dx`
`" "=int((x^(4)-x^(2)+1)+x^(2))/(x^(6)+1)dx`
`" "=int(x^(4)-x^(2)+1)/(x^(6)+1)dx+int(x^(2))/(x^(6)+1)dx`
`" "=int(x^(4)-x^(2)+1)/((x^(2)+1)(x^(4)-x^(2)+1))dx+int(x^(2))/(x^(6)+1)dx`
`rArr" "I=int(1)/(x^(2)+1)dx+int(x^(2))/(x^(6)+1)dx`
माना `" "x^(3)=t`
`rArr" "3x^(2)dx=dt`
`rArr" "x^(2)dx=(dt)/(3)`
`therefore" "I=int(1)/(x^(2)+1)dx+int(dt)/(3(t^(2)-1))`
`" "=tan^(-1)x+(1)/(3)tan^(-1)t+c`
`" "=tan^(-1)x+(1)/(3)tan^(-1)x^(2)+c`