Tìm nguyên hàm của hàm số sau: ∫x3/(x−1)10dx

Question

\(\int \frac{x^{3} }{\left(x-1\right)^{10} } {\rm d}x \)

Answer 1

 Đặt \(t=x-1\Leftrightarrow {\rm d}t={\rm d}x.\)

Ta có: \(\int \frac{x^{3} }{\left(x-1\right)^{10} } {\rm d}x =\int \frac{\left(t+1\right)^{3} }{t^{10} } {\rm d}t =\int \frac{1}{t^{7} } {\rm d}t +\int \frac{3}{t^{8} } {\rm d}t +\int \frac{3}{t^{9} } {\rm d} t+\int \frac{1}{t^{10} } {\rm d}t \)
\(=-\frac{1}{6t^{6} } -\frac{3}{7t^{7} } -\frac{3}{8t^{8} } -\frac{1}{9t^{9} } +C\)
\(=-\frac{1}{6\left(x-1\right)^{6} } -\frac{3}{7\left(x-1\right)^{7} } -\frac{3}{8\left(x-1\right)^{8} } -\frac{1}{9\left(x-1\right)^{9} } +C.\)