intfracx22xdxx12Solution integral 2 xx21x2 dx For the integr

|int|frac(x2+2x)dx(x+1)2

Solution

integral (2 x+x^2)/(1+x)^2 dx For the integrand (x^2+2 x)/(x+1)^2, do long division: = integral (1-1/(x+1)^2) dx Integrate the sum term by term and factor out constants: = integral 1 dx- integral 1/(x+1)^2 dx For the integrand 1/(x+1)^2, substitute u = x+1 and du = dx: = integral 1 dx- integral 1/u^2 du The integral of 1/u^2 is -1/u: = 1/u+ integral 1 dx The integral of 1 is x: = 1/u+x+constant Substitute back for u = x+1: = (x^2+x+1)/(x+1)+constant Which is equal to: = x+1/(x+1)+constant