Question

Evaluate using long division first to write f(x) as the sum of a polynomial and a proper rational function. (Remember to use absolute values where appropriate.) (x3 + 8x2 + 6) dx x + 8

Answer 1

No long division needed here, really, since

$\dfrac{x^3+8x^2+6}{x+8}=\dfrac{x^2(x+8)+6}{x+8}=x^2+\dfrac6{x+8}$

Then the integral is trivial:

$\displaystyle\int\frac{x^3+8x^2+6}{x+8}\,\mathrm dx=\int x^2+\frac6{x+8}\,\mathrm dx=\frac{x^3}3+6\ln|x+8|+C$