Tìm nguyên hàm của hàm số sau: ∫x/x4−3x2+2dx

Question

\(\int \frac{x}{x^{4} -3x^{2} +2} {\rm d}x\)

Answer 1

\(\int \frac{x}{x^{4} -3x^{2} +2} {\rm d}x . \)
Ta có: \(\int \frac{x}{x^{4} -3x^{2} +2} {\rm d}x =\int \frac{x}{\left(x^{2} -2\right)\left(x^{2} -1\right)} {\rm d}x .\)

 Đặt \(t=x^{2} -2\Leftrightarrow {\rm d}t=2x{\rm d}x.\)

 Khi đó: \(\int \frac{x}{x^{4} -3x^{2} +2} {\rm d}x =\int \frac{x}{\left(x^{2} -2\right)\left(x^{2} -1\right)} {\rm d}x =\frac{1}{2} \int \frac{1}{t\left(t+1\right)} {\rm d}t \)
\(=\frac{1}{2} \left[\int \frac{1}{t} {\rm d}t -\int \frac{1}{\left(t+1\right)} {\rm d}t \right]\)

\(=\frac{1}{2} \ln \left|\frac{t}{t+1} \right|+C=\frac{1}{2} \ln \left|\frac{x^{2} -2}{x^{2} -1} \right|+C.\)