evaluate the integral.\n int \frac{16x^{4}+10x^{2}-2x}{x}dx

Explanation:

Step1: Simplify the integrand

First, divide each term in the numerator by $x$: $\frac{16x^{4}+10x^{2}-2x}{x}=16x^{3}+10x - 2$.

Step2: Integrate term - by - term

Use the power rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$). For $\int 16x^{3}dx$, we have $16\times\frac{x^{3 + 1}}{3+1}=4x^{4}$. For $\int 10xdx$, we have $10\times\frac{x^{1+1}}{1 + 1}=5x^{2}$. For $\int(-2)dx=-2x$.

Step3: Combine the results

The integral $\int(16x^{3}+10x - 2)dx=4x^{4}+5x^{2}-2x + C$.

Answer:

$4x^{4}+5x^{2}-2x + C$