Answer to Question #174632 in Calculus for Phyroe

Question

Evaluate the integral of (x³- 2x+5)/(x-3) dx

Answer 1

Convert the expression $\frac{x^3-2x+5}{x-3}$ so that the numerator contains a polynomial of smaller degree than the denominator

$$\frac{x^3-2x+5}{x-3}=\frac{x^3-3x^2+3x^2-9x+7x-21+26}{x-3}=$$

$$=\frac{x^2(x-3)+3x(x-3)+7(x-3)+26}{x-3}=x^2+3x+7+\frac{26}{x-3}$$

Therefore, we can start evaluating the integral,

$$\intop\frac{x^3-2x+5}{x-3}dx=\intop(x^2+3x+7+\frac{26}{x-3})dx=$$

$$=\frac{x^3}{3}+\frac{3x^2}{2}+7x+26 ln|x-3|+C$$