5 - 18 find the general indefinite integral.\n5. $int (x^{2}+x^{-2})dx$

Explanation:

Step1: Use integral sum - rule

$\int(x^{2}+x^{-2})dx=\int x^{2}dx+\int x^{-2}dx$

Step2: Apply power - rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$)

For $\int x^{2}dx$, $n = 2$, so $\int x^{2}dx=\frac{x^{2+1}}{2 + 1}=\frac{x^{3}}{3}$; for $\int x^{-2}dx$, $n=-2$, so $\int x^{-2}dx=\frac{x^{-2 + 1}}{-2+1}=-x^{-1}$

Step3: Combine results and add constant of integration

$\int x^{2}dx+\int x^{-2}dx=\frac{x^{3}}{3}-x^{-1}+C$

Answer:

$\frac{x^{3}}{3}-\frac{1}{x}+C$