经典不定积分一例
题目
求不定积分
\[\int \dfrac{1-x^4}{1+x^6}\mathrm{d}x
\]
解答
\[\begin{split}
\int \dfrac{1-x^4}{1+x^6}\mathrm{d}x
&=\int \dfrac{1-x^2}{x^4-x^2+1}\mathrm{d}x\\
&=\int\frac{x-\frac1x}{(x+\frac{1}{x})^2-3}\dfrac{\mathrm{d}x}{x}\\
\end{split}
\]
熟知
\[\mathrm{d}\left(x+\frac{1}x\right)= \left(\dfrac{x^2-1}{x^2}\right)\mathrm{d}x=\left(\dfrac{x-\frac{1}x}{x}\right)\mathrm{d}x
\]
作换元 \(t=x+\frac1x\)
\[\begin{split}
\int \dfrac{1-x^4}{1+x^6}\mathrm{d}x
&=\int \dfrac{\mathrm{d}t}{t^2-3}\\
&=\int \dfrac{\left(t+\sqrt3\right)\mathrm{d}t-\left(t-\sqrt3\right)\mathrm{d}t}{2\sqrt3\left(t-\sqrt3\right)\left(t+\sqrt3\right)} \\
&=\frac{1}{2\sqrt3}\left(\int\frac{\mathrm{d}t}{t-\sqrt3}-\int\frac{\mathrm{d}t}{t+\sqrt3}\right)\\
&=\frac{\ln\left\vert\frac{t-\sqrt{3}}{t+\sqrt{3}}\right\vert}{2\sqrt3} + C\\
&=\frac{\ln\left(\frac{x^2-\sqrt{3}x+1}{x^2+\sqrt{3}x+1}\right)}{2\sqrt3} + C
\end{split}
\]
