(a)OR(b)

Find : \[\int \frac{\mathrm{x}^{2}}{\left(\mathrm{x}^{2}+4\right)\left(\mathrm{x}^{2}+9\right)} d \mathrm{x} \]

$\displaystyle \int \frac{\mathrm{x}^2}{(\mathrm{x}^2+4)(\mathrm{x}^2+9)}\,dx$. With $\displaystyle t=\mathrm{x}^2$, $\displaystyle \frac{t}{(t+4)(t+9)}=\frac{-4/5}{t+4}+\frac{9/5}{t+9}$, so the integrand $\displaystyle =-\frac45\cdot\frac{1}{\mathrm{x}^2+4}+\frac95\cdot\frac{1}{\mathrm{x}^2+9}$. Thus the integral $\displaystyle =-\frac{2}{5}\tan^{-1}\frac{\mathrm{x}}{2}+\frac{3}{5}\tan^{-1}\frac{\mathrm{x}}{3}+C$.