determine the following indefinite integral.\n int 5m(12m^{3}-5m)dm \n int 5m(12m^{3}-5m)dm=square

Explanation:

Step1: Expand the integrand

First, expand $5m(12m^{3}-5m)$ using the distributive - property: $5m\times12m^{3}-5m\times5m = 60m^{4}-25m^{2}$. So the integral becomes $\int(60m^{4}-25m^{2})dm$.

Step2: Apply the power - rule for integration

The power - rule for integration is $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$). For $\int60m^{4}dm$, we have $60\times\frac{m^{4 + 1}}{4+1}=12m^{5}$. For $\int-25m^{2}dm$, we have $-25\times\frac{m^{2 + 1}}{2+1}=-\frac{25}{3}m^{3}$.

Step3: Combine the results and add the constant of integration

The indefinite integral $\int(60m^{4}-25m^{2})dm=12m^{5}-\frac{25}{3}m^{3}+C$.

Answer:

$12m^{5}-\frac{25}{3}m^{3}+C$