∫((1-x+2x^2-x^3)/(x(x^2+1)^2) dx Integrate by partial fractions

Question

I keep coming up one term short in my answer.  Will someone please integrate this problem using the partial fraction method, clearly showing the steps?

Answer 1

to find

we can use Fraction method

then

Apply formula:

so

 

Answer 2

Partial frations of

Equate the terms in both sides we get,

A = 1                    -> (1)

C + E = -1           -> (2)

2A + B + D = 2    -> (3)

C = -1                  -> (4)

A + B = 0            -> (5)

Substitute (1) in (5) we get,

A + B = 0

1 + B = 0

 B = -1

Substitute (4) in (2) we get,

C + E = -1

-1 + E = -1

E = -1 + 1 = 0

Substitute A and B values in (3) we get,

2A + B + D = 2

2(1) + (-1) + D = 2

2 - 1 + D = 2

1 + D = 2

D = 2 -1 = 1

Therefore A = 1, B = -1, C = -1, D = 1, E = 0

Substitute A, B,C,D and E values in the partial equation we get,

Now do the integration.