Question

Solve: $\int {\left(x+\frac{1}{x}\right)}^{3}dx$

Answer 1

$\begin{array}{c}I=\int {\left(x+\frac{1}{x}\right)}^{3}dx\\ =\int \left(x^3+\frac{1}{x^3}+3x^2\times \frac{1}{x}+3x\times {\left(\frac{1}{x}\right)}^2\right)dx\\ =\int \left(x^3+\frac{1}{x^3}+3x+\frac{3}{x}\right)dx\\ =\frac{x^4}{4}-\frac{1}{2x^2}+\frac{3x^2}{2}+3\log x+c\end{array}$