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Integrate the function: 1x-x3

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Question

Integrate the function:

`1/(x - x^3)`

Sum
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Solution

Let `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)

Putting x = 0 in equation (1),

1 = A(1 + 0) (1 – 0)

⇒ A = 1

Putting x = -1 in equation (1),

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

`=> B = - 1/2`

Putting x = 1 in equation (1),

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

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Chapter 7: Integrals - Exercise 7.12 [Page 352]

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NCERT Mathematics Part 1 and 2 [English] Class 12
Chapter 7 Integrals
Exercise 7.12 | Q 1 | Page 352

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