question evaluate the indefinite integral given below. ∫((x^6 - 5x)/x^2)dx provide your answer below: ∫((x^6…

Explanation:

Step1: Simplify the integrand

$\frac{x^{6}-5x}{x^{2}}=\frac{x^{6}}{x^{2}}-\frac{5x}{x^{2}}=x^{4}-\frac{5}{x}$

Step2: Integrate term - by - term

$\int(x^{4}-\frac{5}{x})dx=\int x^{4}dx - 5\int\frac{1}{x}dx$ Using the power rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$) and $\int\frac{1}{x}dx=\ln|x|+C$, we have $\int x^{4}dx=\frac{x^{5}}{5}+C_1$ and $5\int\frac{1}{x}dx = 5\ln|x|+C_2$.

Step3: Combine the results

$\int(x^{4}-\frac{5}{x})dx=\frac{x^{5}}{5}-5\ln|x|+C$ (where $C = C_1 - C_2$)

Answer:

$\frac{x^{5}}{5}-5\ln|x|+C$