Tìm nguyên hàm của hàm số sau: ∫2x+1/x4+2x3+3x2+2x−3dx

Question

\(\int \frac{2x+1}{x^{4} +2x^{3} +3x^{2} +2x-3} {\rm d}x\)

Answer 1

Đặt \(t=x^{2} +x\Rightarrow {\rm d}t=\left(2x+1\right){\rm d}x\)Khi đó:
\(\begin{array}{l} {\int \frac{2x+1}{x^{4} +2x^{3} +3x^{2} +2x-3} {\rm d}x=\int \frac{2x+1}{\left(x^{2} +x\right)^{2} +2\left(x^{2} +x\right)-3} {\rm d}x=\int \frac{{\rm d}t}{t^{2} +2t-3} =\int \frac{1}{\left(t-1\right)\left(t+3\right)} {\rm d}t    } \\ {\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, =\frac{1}{4} \int \left(\frac{1}{t-1} -\frac{1}{t+3} \right){\rm d}t =\frac{1}{4} \ln \frac{t-1}{t+3} +C=\frac{1}{4} \ln \frac{x^{2} +x-1}{x^{2} +x+3} +C } \end{array}\)