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

Explanation

माना `I=int(1)/(x^(4)+1)dx`
`=(1)/(2)int(2)/(x^(4)+1)dx=(1)/(2)int((x^(2)+1)-(x^(2)-1))/(x^(4)+1)dx`
`=(1)/(2)int(x^(2)+1)/(x^(4)+1)dx-(1)/(2)int(x^(2)-1)/(x^(4)+1)dx`
`=(1)/(2)int(1+(1)/(x^(2)))/(x^(2)+(1)/(x^(2)))dx-(1)/(2)int(1-(1)/(x^(2)))/(x^(2)+(1)/(x^(2)))dx`
`=(1)/(2)int(1+(1)/(x^(2)))/((x-(1)/(x))^(2)+2)dx-(1)/(2)int(1-(1)/(x^(2)))/((x+(1)/(x))^(2)-2)dx`
अब, प्रथम समाकलन में `x-(1)/(x)=u` अर्थात `(1+(1)/(x^(2)))dx=du`
तथा द्वितीय समाकलन में `x+(1)/(x)=v` अर्थात `(1-(1)/(x^(2)))dx=dv` रखने पर
`I=(1)/(2)int(du)/(u^(2)+(sqrt2)^(2))-(1)/(2)int(dv)/(v^(2)-(sqrt2)^(2))`
`=(1)/(2).(1)/(sqrt2)tan^(-1)((u)/(sqrt2))-(1)/(2).(1)/(2sqrt2)log((v-sqrt2)/(v+sqrt2))+c`
`=(1)/(2sqrt2)tan^(-1)((x-1//x)/(sqrt2))-(1)/(4sqrt2)log((x+1//x-sqrt2)/(x+1//x+sqrt2))+c`
`=(1)/(2sqrt2)tan^(-1)((x^(2)-1)/(sqrt2x))-(1)/(4sqrt2)log((x^(2)-sqrt2x+1)/(x^(2)+sqrt2x+1))+c`