Problem 28 - Calculus Early Transcendentals

Question

Determine the following indefinite integrals. Check your work by differentiation. $$\int 5 m\left(12 m^{3}-10 m\right) d m$$

Step-by-Step Solution

Answer

Answer: The indefinite integral of the function \(5m(12m^3 - 10m)\) with respect to m is \(12m^5 - \frac{50}{3}m^3 + C\).

Step 1: Simplify the integrand

First, we need to simplify the integrand. We can do this by distributing the 5m inside the parentheses: $$5m\left(12m^3 - 10m\right) = 60m^4 - 50m^2$$

Step 2: Find the indefinite integral

Next, we'll find the indefinite integral of the simplified integrand with respect to m: $$\int \left(60m^4 - 50m^2\right) dm$$ We can apply power rule of integration: $$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$ Applying the rule to each term gives: $$\int 60m^4 dm - \int 50m^2 dm = \frac{60m^5}{5} - \frac{50m^3}{3} + C$$ This simplifies to: $$12m^5 - \frac{50}{3}m^3 + C$$

Step 3: Differentiate the result

Now we need to differentiate the result obtained in step 2 to check if it matches the original integrand, which is \(60m^4 - 50m^2\). Using power rule of differentiation: $$\frac{d}{dx}\left(x^n\right) = nx^{n-1}$$ Differetiating the result we have: $$\frac{d}{dm}\left(12m^5 - \frac{50}{3}m^3 + C\right) = 60m^4 - 50m^2$$ Since the differentiated result matches the original integrand, our indefinite integral is correct. So, the final answer is: $$\int 5 m\left(12 m^{3}-10 m\right) d m = 12m^5 - \frac{50}{3}m^3 + C$$