Problem 12 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the indefinite integral. $$\int 2 x^{5} d x$$

Step-by-Step Solution

Answer

The indefinite integral of \(2x^5\) is \(\frac{x^6}{3} + C\).

Step 1: Identify the function and its power

We are given the function $$2x^5$$. The function is a power function with a coefficient, where the power, n, is 5 and the coefficient is 2.

Step 2: Apply the power rule

Now that we have identified the function and its power, we can apply the power rule: $$\int 2x^{5} dx = 2\int x^{5} dx = 2\left(\frac{x^{5+1}}{5+1}\right) + C$$ This simplifies to: $$2\left(\frac{x^6}{6}\right) + C$$

Step 3: Simplify the result

Now we can simplify the expression by multiplying the coefficient (2) by the fraction: $$\frac{2x^6}{6} + C$$ The 2 and 6 have a common factor of 2, which we can cancel out to get the final answer: $$\frac{x^6}{3} + C$$ Thus, the indefinite integral of $$\int 2x^5 dx$$ is $$\frac{x^6}{3} + C$$.

Key Concepts

Power Rule in Integration

The Power Rule makes evaluating indefinite integrals straightforward, especially when dealing with polynomial expressions. It states that for any function in the form of \( x^n \), with \( n eq -1 \), the integral is given by:

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)

In the original exercise, we used the Power Rule to tackle the term \( 2x^5 \).
First, isolate the constant from the power function, which allows you to separately multiply it back after integration:

• \( 2 \int x^5 \, dx = 2 \left( \frac{x^{5+1}}{5+1} \right) + C \)

This results in \( 2 \left( \frac{x^6}{6} \right) \), where integration has been completed using the Power Rule.
The simplicity of the Power Rule allows even more complex polynomial expressions to be integrated easily by applying this rule term by term.

Mathematical Simplification

After applying the Power Rule, we focus on simplifying the resulting expression to its simplest form. This simplification is important for making results more interpretable and manageable.
When we initially perform the integration, we arrive at the expression \( \frac{2x^6}{6} + C \).
Here, observe the common factor between the numerator and the denominator: both can be divided by 2.

• Reducing the fraction \( \frac{2}{6} \) gives \( \frac{1}{3} \).

When simplified, the expression becomes \( \frac{x^6}{3} + C \).
This process emphasizes the importance of looking for any common factors after integration to achieve the most reduced form of the solution.

Integration Constants

Whenever an indefinite integral is found, a constant \( C \) is added. This constant represents the family of all antiderivatives of a function because differentiation loses "information" about constants.
Every indefinite integral involves an integration constant because:

• The operation of differentiation of a constant is zero, meaning the original function could include any constant value.
• Without the constant, you only represent one particular antiderivative instead of the entire family.

In the solution of our exercise, the result was \( \frac{x^6}{3} \), but to account for all possible antiderivatives, we include \( + C \).
Acknowledging the integration constant helps solidify understanding that the integral provides not just a single solution but rather a set of potential solutions.