Đặt: \(\left\{\begin{array}{l}u=x \\ d v=f^{\prime}(x) d x\end{array} \Rightarrow\left\{\begin{array}{l}d u=d x \\ v=f(x)\end{array}\right.\right.\)

Ta có: \(\int_{0}^{1} x f^{\prime}(x) d x=x .\left.f(x)\right|_{0} ^{1}-\int_{0}^{1} f(x) d x=f(1)-\int_{0}^{1} f(x) d x\).

Theo già thiết: \(\int_{0}^{1} x f^{\prime}(x) d x=\frac{1}{36}, f(1)=-\frac{1}{18}\)

\(\Rightarrow-\frac{1}{18}-\int_{0}^{1} f(x) d x=\frac{1}{36} \Rightarrow \int_{0}^{1} f(x) dx=-\frac{1}{18}-\frac{1}{36}=-\frac{1}{12}\)