1.

फलन `int(x^(2)-1)/(x^(4)+x^(2)+1)dx` का x के सापेक्ष समाकलन कीजिए ।

Answer» माना `" "I=int(x^(2)-1)/(x^(4)+x^(2)+1)dx`
अंश व हर `x^(2)` को से भाग देने पर, (
`I=int((1-1//x^(2))dx)/(x^(2)+1//x^(2)+1)=int((1-1//x^(2))dx)/((x^(2)+2+1//x^(2)-1)`
`" "` (नोट कीजिए )
`=int((1-1//x^(2))dx)/((x+1//x)^(2)-1)`
माना `x+1//x=t rArr (1-1//x^(2))dx=dt`
`therefore " "I=int(dt)/(t^(2)-1)=int(dt)/((t-1)(t+1))`
`=(1)/(2)[int(dt)/(t-1)-int(dt)/(t+1)]`
`=(1)/(2)[int(dt)/(t-1)-int(dt)/(t+1)]`
`=(1)/(2)log((t-1)/(t+1))=(1)/(2)log[(x+(1)/(x)-1)/(x+(1)/(x)+1)]`
`=(1)/(2)log[(x^(2)-x+1)/(x^(2)+x+1)]`


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