What is primitive function of `f(x)=1/(x(x^2+1))` ?

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llltkl | College Teacher | (Level 3) Valedictorian

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To find the primitive function of f(x) we have to find the indefinite integral of the given function.

First we need to split the given function using partial fractions,

`1/(x(x^2 + 1))= A/x + (Bx + C)/(x^2 + 1)`

`rArr 1 = A(x^2 + 1) + x(Bx + C)` ...............(i)

Substituting x = 0 into equation (i)

`1 = A + 0`

`rArr A=1`

Again substituting x = 1 into the same equation we get:

`1 = 2A + B + C`

Since `A = 1` ,

`1 = 2 + B + C`

`rArr B + C = -1` ...........(ii)

Substituting x = -1,

`1 = 2A + B – C`

`rArr B - C = -1` ..........(iii)

Adding equation (ii) and (iii) together,

`2B = -2`

`rArr B = -1`

So,  `A = 1, B = -1, C = 0.`

Putting these values back into our original equation we get:

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

Now, `int( 1/x - x/(x^2 + 1))dx`

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

`= ln(x) - (1/2) * ln(x^2 + 1) +c`

`rarr` answer.