Question

निम्नलिखित फलन का समाकलन कीजिए।

`1/(x - x^3)`

Answer 1

माना `1/(x - x^3) = 1/(x(1 + x)(1 - x))`

`≡ A/x + B/(1 + x) + C/(1 - x)`

⇒ 1 = A (1 + x) (1 – x) + Bx (1 – x) + Cx (1 + x)    …(1)

समीकरण (1) में, x = 0 रखने पर,

1 = A(1 + 0) (1 – 0)

⇒ A = 1

समीकरण (1) में, x = -1 रखने पर,

1 = B (-1) (1 + 1)

`=> B = - 1/2`

समीकरण (1) में, x = 1 रखने पर,

1 = C(1)(1 + 1)

`=> C = 1/2`

`therefore 1/(x - x^3) = 1/x - 1/(2(1 + x)) + 1/(2(1 - x))`

`therefore int 1/(x - x^3) dx = int 1/x dx - 1/2 int 1/(1 + x) dx + 1/2 int 1/(1 - x) dx`

`= log |x| - 1/2 log |1 + x| - 1/2 log |1 - x| + C`

`= 1/2 log |x|^2 - 1/2 log |1 - x^2| + C`

`= 1/2 log |x^2/(1 - x^2)|` +  C