determine the following indefinite integral.\n int\frac{4x^{7}+6x^{5}}{x^{4}}dx \n int\frac{4x^{7}+6x^{5}}{x^…

Explanation:

Step1: Simplify the integrand

Divide each term in the numerator by $x^{4}$: $\frac{4x^{7}+6x^{5}}{x^{4}}=\frac{4x^{7}}{x^{4}}+\frac{6x^{5}}{x^{4}} = 4x^{7 - 4}+6x^{5 - 4}=4x^{3}+6x$.

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 $\int4x^{3}dx$, we have $4\times\frac{x^{3 + 1}}{3+1}=x^{4}$. For $\int6xdx$, we have $6\times\frac{x^{1+1}}{1 + 1}=3x^{2}$. So $\int(4x^{3}+6x)dx=x^{4}+3x^{2}+C$.

Answer:

$x^{4}+3x^{2}+C$