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

Answer

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Explanation

`(1)/(1+x^(3))=(1)/((1+x)(1-x+x^(2)))=A/(1+x)+(Bx+C)/(1-x+x^(2))`
हल करने पर
`(1)/(1+x^(3))=1/3 1/(1+x)+(2-x)/(3(1-x+x^(2)))`
`:. int 1/(1+x^(3)) dx=1/3 int (dx)/(1+x) +1/3 int ((2-x) dx)/(1-x+x^(2))`
`=1/3 log_(e) |1+x|+1/2 . 1/3 int ((4-2x)dx)/(1-x+x^(2))`
`=1/3 log_(e) |1+x|+1/6 int (3-(2x-1))/(1-x+x^(2)) dx`
`=1/3 log_(e) |1+x|+1/2 int (dx)/(1-x+x^(2))-1/6 int (2x-1)/(1-x+x^(2)) dx`
`=1/3 log_(e) |1+x|+1/2 int (dx)/((x^(2)-x+1/4)+3/4) -1/6 int (2x-1)/(1-x+x^(2)) dx`
`=1/3 log_(e) |1+x|+1/2 int (dx)/((x-1/2)^(2)+(sqrt(3)//2)^(2))-1/6 log|1+x+x^(2)|dx`
`=1/6 log_()|1+x|+1/2 1/(sqrt(3)//2) "tan"^(-1) (x-1/2)/(sqrt(3)//2) - 1/6 log |1-x+x^(2)|+c`
`=1/6 log_(e) |((1+x)^(2))/(1-x+x^(2))|+1/sqrt(3) "tan"^(-1) (2x-1)/sqrt(3)+c`.