Diện tích hình phẳng cần tìm là:
$S=\int\limits_1^2|\dfrac{1}{x(x^3+1)}|dx$
$=\int\limits_1^2\dfrac{1}{x(x^3+1)}dx$
$=\int\limits_1^2\left(\dfrac{1}{x}-\dfrac{x^2}{x^3+1}\right)dx$
$=\int\limits_1^2\dfrac{dx}{x}-\dfrac{1}{3}\int\limits_1^2\dfrac{d(x^3+1)}{x^3+1}$
$=\ln x\left|\begin{array}{l}2\\1\end{array}\right.-\dfrac{1}{3}\ln(x^3+1)\left|\begin{array}{l}2\\1\end{array}\right.$
$=\dfrac{2}{3}\ln\dfrac{4}{3}$