Question

Evaluate $\int \frac{2x^2+3}{x^2+5x+6}dx$

Answer 1

$\int \frac{2x^2+3}{x^2+5x+6}dx$

$=\int \frac{2x^2+3}{x^2+2x+3x+6}dx$

$=\int \frac{2x^2+3}{x(x+2)+3(x+2)}dx$

$=\int \frac{2x^2+3}{(x+2)(x+3)}dx$

Solving by the method of partial fractions,

$\frac{2x^2+3}{(x+2)(x+3)}=\frac{A}{x+2}+\frac{B}{x+3}$

$\Rightarrow 2x^2+3=A(x+3)+B(x+2)$

Put $x=-3\Rightarrow 18+3=-B\therefore B=-21$

Put $x=-2\Rightarrow 8+3=A\therefore A=11$

$\int \frac{2x^2+3}{(x+2)(x+3)}dx=11\int \frac{dx}{x+2}-21\int \frac{dx}{x+3}$

$=11\log |x+2|-21\log |x+3|+c$