Question

partial fraction of x^2+x+1/X+2(x^2+1)​

Answer 1

Answer:

Input:

x^2 + x + 1/X + 2 (x^2 + 1)

3D plot:

3D plot

Contour plot:

Contour plot

Alternate form:

(3 x^2 X + x X + 2 X + 1)/X

Expanded form:

3 x^2 + x + 1/X + 2

Roots:

3 x^2 + x + 2!=0, X = 1/(-3 x^2 - x - 2)

X = 1/(-3 x^2 - x - 2)

Properties as a real function:

Domain

R (all real numbers)

Range

{y element R : X!=0 and X (X (12 y - 23) - 12)>=0}

Roots for the variable x:

x = 1/6 (-sqrt(-23 X - 12)/sqrt(X) - 1)

x = 1/6 (sqrt(-23 X - 12)/sqrt(X) - 1)

Derivative:

d/dx(x^2 + x + 1/X + 2 (x^2 + 1)) = 6 x + 1

Indefinite integral:

integral(x + x^2 + 2 (1 + x^2) + 1/X) dx = x^3 + x^2/2 + x/X + 2 x + constant

Series representations:

x + x^2 + 2 (1 + x^2) + 1/X = 3 + x + 3 x^2 + sum_(n=1)^∞ (-1)^n (-1 + X)^n

x + x^2 + 2 (1 + x^2) + 1/X = sum_(n=-∞)^∞ ( piecewise | 1 | n = -1

2 + x + 3 x^2 | n = 0) X^n

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