Đặt \(t=1+x^{4} \Rightarrow {\rm d}t=4x^{3} {\rm d}x\Rightarrow x^{3} {\rm d}x=\frac{{\rm d}t}{4} \)
Khi đó: \(\int \frac{x^{7} }{\left(1+x^{4} \right)^{2} } {\rm d}x=\int \frac{x^{4} .x^{3} }{\left(1+x^{4} \right)^{2} } {\rm d}x=\int \frac{t-1}{t^{2} } \frac{{\rm d}t}{4} \)
\(=\frac{1}{4} \int \left(\frac{1}{t} -\frac{1}{t^{2} } \right){\rm d}t=\frac{1}{4} \left(\ln \left|t\right|+\frac{1}{t} \right)+C=\frac{1}{4} \left(\ln \left(1+x^{4} \right)+\frac{1}{1+x^{4} } \right)+C \)