Question

Evaluate Integrate {(1-x³)/x+1}dx

Answer 1

`-((x+1)^3)/(3)+(3)/(2)(x+1)^2-3(x+1)-2\ln |x+1|+C`

Explanation:

Let
`t=x+1`. Then,
`x=t-1` and
`dx=dt`.

`\int (1-x^3)/(x+1) dx=\int (1-(t-1)^3)/(t) dt \\ \\ =\int (-t^3+3t^2-3t+2)/(t) dt \\ \\ =\int (-t^2+3t-3+2t^(-1)) dt \\ \\ =-(t^3)/(3)+(3)/(2)t^2-3t-2\ln |t|+C \\ \\ =-((x+1)^3)/(3)+(3)/(2)(x+1)^2-3(x+1)-2\ln |x+1|+C`