`int (x^(2)+1)/((x+1)^(3)(x-2)) dx` का मान ज्ञात कीजिए।

Answer

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Explanation

मानलो `x+1=y`
`:. (x^(2)+1)/((x+1)^(3)(x-2))=((y-1)^(2)+1)/(y^(3)(y-1-2))=(y^(2)-2y+2)/(y^(3)(y-3))`
`=1/y^(3) ((2-2y+y^(2))/(-3+y))` [y में घातों को आरोही क्रम में रखने पर]
`=1/y^(3) [-2/3+4/9 y-5/27 y^(2) +(5y^(3))/(27(-3+y))]` (भाग देने पर)
`=-2/(3y^(3))+4/(9y^(2))-5/(27y)+5/(27(-3+y))`
`=- 2/(3(x+1))+4/(9(x+1)^(2))-5/(27(x+1))+5/(27(x-2))`
`=-2/3 (x+1)^(-3) +4/9 (x+1)^(-2) -5/27 1/(x+1)+5/27 1/(x-2)`
`:. int (x^(2)+1)/((x+1)^(3)(x-2)) dx=-2/3 int (x+1)^(-3) dx+4/9 int (x+1)^(-2) dx - 5/27 int (dx)/(x+1)+5/27 int (dx)/(x-2)`
`=2/3 ((x+1)^(-2))/(-2)+4/9 ((x+1)^(-1))/(-1) -5/27 log |x+1|+5/27 log |x-2|`
`=5/27 [log_(e) |x-2|-log_(e) |x+1|]+1/3 (x+1)^(-2)-4/9 (x+1)^(-1)`
`=5/27 log_(e) |(x-2)/(x+1)|+1/3 (x+1)^(-2) -4/9 (x+1)^(-1) +c`.