问题及解答

求不定积分

\[\int\frac{1}{x(x^6+4)}dx.\]

\[
\begin{split}
\int\frac{1}{x(x^6+4)}dx&=\int\frac{x^5}{x^6(x^6+4)}dx\\
&=\frac{1}{6}\int\frac{dx^6}{x^6(x^6+4)}\\
&\stackrel{t=x^6}{=}\frac{1}{6}\int\frac{dt}{t(t+4)}\\
&=\frac{1}{6}\cdot\frac{1}{4}\int(\frac{1}{t}-\frac{1}{t+4})dt\\
&=\frac{1}{24}(\ln|t|-\ln|t+4|)+C\\
&=\frac{1}{24}(\ln|x^6|-\ln|x^6+4|)+C.
\end{split}
\]