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Evaluate Integrate {(1-x³)/x+1}dx

1 Answer

1 vote

Answer:


-((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

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