Ta có \(I=\int_{0}^{1} x^{3} f^{\prime}\left(x^{2}\right) d x=\int_{0}^{1} x^{2} f^{\prime}\left(x^{2}\right) x d x\)

Đặt \(t=x^{2} \Rightarrow d t=2 x d x \Leftrightarrow \frac{1}{2} d t=x d x\)

Đổi cận: \(x=0 \Rightarrow t=0\)

Suy ra: \(I=\frac{1}{2} \int_{0}^{1} t f^{\prime}(t) d t\)

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

Khi đó: \(I=\frac{1}{2}\left(\left.t f(t)\right|_{0} ^{1}-\int_{0}^{1} f(t) d t\right)=\frac{1}{2}\left(f(1)-\int_{0}^{1} f(x) d x\right)=\frac{1}{2}(4-3)=\frac{1}{2}\).