Integrate $ \int \frac{x^2}{x+1} dx $

Question

How can I integrate by changing variable or by parts?

$$ \int \frac{x^2}{x+1} dx $$

Thanks a lot

Answer 1

Hint: $$\frac{x^2}{x+1} = x-\frac{x}{x+1}$$ If you are integrating a ratio of polynomials where the numerator is of higher degree than the denominator, always try polynomial division.

Answer 2

Changing variables as you ask, but there are better ways

Take $x=y-1$ so $\int \frac{x^2}{x+1} dx=\int \frac{y^2-2y+1}{y} dy=\int y dy + \int -2 dy + \int \frac{1}{y}$ so $\int \frac{x^2}{x+1} dx=\frac{y^2}{2}-2y+log(y)$ Subsitue again and you are done.