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

Explanation

माना `I = (dx)/(sin^(4) x+cos^(4)x)`
अश व हर को `cos^(4)x` से भाग देंने पर
`I =int(sec^(4)xdx)/(1+tan^(4)x)`
`= int((1+tan^(2)x)sec^(2)xdx)/(1+tan^(4)x)`
माना `x = t therefore" "sec^(2) xdx = dt`
`therefore" "I int((1+t^(2))/(1+t^(4)))dt`
अश व हर को `t^(2)` से भाग देने पर
`I = int(1+1//t^(2))/(t^(2) + 1//t^(2))dt`
माना `t - (1)/(t) =urArr (1+(1)/(t^(2)))dt= du`
पुन: `t^(2) +(1)/(t^(2)) = t^(2) +(1)/(t^(2)) -2+2`
` = (t -(1)/(t))^(2) +2 = u^(2) + 2`
`therefore I = int(du)/(2+u^(2))= (1)/(sqrt(2))tan^(-1)((u)/(sqrt(2)))`
` = (1)/(sqrt(2))tan^(-1)[(1)/(sqrt(2))(t-(1)/(t))] = (1)/(sqrt(2))tan^(-1).((t^(2)-1)/(tsqrt(2)))`
` = (1)/(sqrt(2))^(-1)((tan^(2)x-1)/(sqrt(2)tanx))`