Question

Type 3. The denominator contains an
theredueible quadradie factors
8. Resolve : x²/(x^2+1)(x+1) into partial freaction
Soln​

Answer 1

Step-by-step explanation:

the forward of formula is over the addition and subtraction the formula later is not ok

Answer 2

Answer:

$\frac{(x - 1)}{2( {x}^{2} + 1) } + \frac{1}{2(x + 1)}$

Step-by-step Explanation:

Let,

$\frac{ {x}^{2} }{( {x}^{2} + 1)(x + 1)} = \frac{ax + b}{ {x}^{2} + 1 } + \frac{c}{x + 1}$

$= > \frac{ {x}^{2} }{( {x}^{2} + 1)(x + 1)} = \frac{(ax + b)(x + 1) + c( {x}^{2} + 1)}{( {x}^{2} + 1)(x + 1) }$

since, the denominator on both side is equal,

therefore, the numerator must be equal

$= > {x}^{2} = (a + c) {x}^{2} + (a + b)x + (b + c)$

Now, comparing tge coefficients of variables,

we get,

a + c = 1 ........(i)

a + b = 0 ........(ii)

b + c = 0 .........(iii)

On subtracting (iii) from (ii) ,

we get,

a - c = 0 ..........(iv)

Now, adding (i) and (iv),

we get,

2a = 1

=> a = ½

therefore,

b = ½ and c = -½

Putting the values of a, b and c in the supposed term,

we get, partial fraction equals to

$\frac{(x - 1)}{2( {x}^{2} + 1) } + \frac{1}{2(x + 1)}$