How Do You Solve Integrals Using Partial Fractions?

[nameVoid]

$$\int \frac{5x^2+11x+17}{x^3+5x^2+4x+20}dx$$
$$\int \frac{5x^2+11x+17}{(x^2+4)(x+5)}dx$$
$$\frac{Ax+B}{x^2+4}+\frac{C}{x+5}=\frac{5x^2+11x+17}{(x^2+4)(x+5)}$$
$$(Ax+B)(x+5)+C(x^2+4)=5x^2+11x+17$$
$$Ax^2+5Ax+Bx+5B+Cx^2+4C=5x^2+11x+17$$
$$x^2(A+C)+x(5A+B)+(5B+4C)=5x^2+11x+17$$
$$A+C=5, 5A+B=11, 5B+4C=17$$
$$A=5-C$$
$$5(5-C)+B=11, 25-5C+B=11, B=-14+5C$$
$$5(-14+5C)+4C=17, -70+29C=17, C=3, B=1, A=2$$
$$\int \frac{2x+1}{x^2+4}+ \frac{3}{x+5}dx$$
$$ln(x^2+4) +aractan(x/2)/2+3ln|x+5|+Z$$
orginally I thought I had made a mistake somwhere but I believe this is correct please make suggestions I am new to this technique

 

[Mark44]

Looks good, but I haven't checked each detail. Two things you can do are
1) check that (2x + 1)/(x² + 4) + 3/(x + 5) = your original integrand.
2) check that d/dx[ln(x² + 4) + 1/2*arctan(x/2) + ln|x + 5| = your original integrand.