Calculate the antiderivative of the function f(x) given by f(x)=2x/(x+1)*(x^2+1)?

We'll have to determine the indefinite integral of the given
function.

For the beginning, we'll re-write the function as an
algebraic sum of elementary fractions:

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

2x = Ax^2 + A + Bx^2 + Bx + Cx +
C

2x = x^2(A + B) + x(B+C) +
A+C

Comparing, we'll get:

A+B = 0
> A = -B (1)

B+C = 2 (2)

A+C = 0
<=>-B + C = 0 (3)

(2)+(3) => 2C = 2 =>
C=1 => A=-1=>B =1

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

Int f(x)dx = Int-dx/(x+1) + Int xdx/(x^2 + 1) + Int
dx/(x^2 + 1)

Int-dx/(x+1) = -ln|x+1| +
C

Int xdx/(x^2 + 1) = Int dt/2t = ln|t|/2 +
C

x^2 + 1 = t

2xdx =
dt

xdx = dt/2

Int dx/(x^2 + 1) = arctan
x + C

Int f(x)dx = -ln|x+1| + ln|t|/2 + arctan x +
C

The antiderivative of the given function is: Int
f(x)dx = -ln|x+1| + ln|t|/2 + arctan x + C.