Problem 44 - Calculus for Biology and Medicine

Question

Compute the indefinite integrals. $$ \int\left(\frac{1}{2} x^{5}+2 x^{3}-1\right) d x $$

Step-by-Step Solution

Answer

\( \frac{1}{12} x^6 + \frac{1}{2} x^4 - x + C \)

Step 1: Identify the Function Components

The given integral is \( \int \left( \frac{1}{2} x^{5}+2 x^{3}-1 \right) dx \). There are three terms within the integrand: \( \frac{1}{2} x^5 \), \( 2x^3 \), and \( -1 \). We will integrate each term separately.

Step 2: Use the Power Rule for Integration

Apply the power rule of integration \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) to each term. The power rule can be applied directly to \( \frac{1}{2} x^5 \), \( 2x^3 \), and \( -1 \).

Step 3: Integrate the First Term

For the first term \( \int \frac{1}{2} x^5 \, dx \), apply the power rule: \( \frac{1}{2} \cdot \frac{x^{6}}{6} = \frac{1}{12} x^6 \).

Step 4: Integrate the Second Term

For the second term \( \int 2x^3 \, dx \), apply the power rule: \( 2 \cdot \frac{x^{4}}{4} = \frac{1}{2} x^4 \).

Step 5: Integrate the Third Term

For the constant \(-1\), the integral is \( \int (-1) dx = -x \).

Step 6: Combine the Results and Add the Constant of Integration

Sum the integrated terms: \( \frac{1}{12} x^6 + \frac{1}{2} x^4 - x + C \), where \( C \) is the constant of integration.

Key Concepts

Understanding the Power Rule

The power rule is a fundamental technique for finding the indefinite integral of a function that has the form \( x^n \). To apply it, you simply increase the exponent of the term by one and then divide the term by the new exponent. This is represented mathematically as: - \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) - \( n \) must not be equal to \(-1\) to avoid division by zero.Let's break this down with an example from the exercise: if you have the term \( \frac{1}{2}x^5 \), applying the power rule means you increase the exponent from \( 5 \) to \( 6 \), resulting in \( \frac{1}{2} \times \frac{x^6}{6} \). It's important to multiply by the original coefficient, which in this case is \( \frac{1}{2} \), so you end up with \( \frac{1}{12}x^6 \). Using the power rule helps transform complex polynomial integrals into simpler, easily computable forms.

The Art of Integration Techniques

Integration involves a variety of techniques, but one of the most straightforward is separating and integrating each term within an integral individually. This is easily used when the integrand is a polynomial. Each term like we saw in the given exercise: \( \frac{1}{2}x^5 + 2x^3 - 1 \), is addressed independently using a technique like the power rule. - Start by identifying each term - Apply relevant integration methods to each term - Sum the results for your final integral For constants such as \(-1\) in our problem, remember that integrating a constant \(c\) results in \( cx \). So, \( \int (-1) \, dx = -x \). This process may include multiple integration techniques, optimizing the approach suitable for each term's complexity. By mastering these steps, integrating becomes a more structured, step-by-step activity rather than a daunting task.

Exploring the Constant of Integration

When calculating indefinite integrals, you will always see a \( + C \) in the solution, representing the constant of integration. But why is it important?Indefinite integration essentially reverses differentiation. When you differentiate a function, constant terms disappear. Hence, when integrating, we must include an arbitrary constant \( C \) in our result to account for any constant that may have vanished during differentiation. For instance, if two functions \( f(x) = x^2 + 3 \) and \( g(x) = x^2 + 5 \) were both differentiated, both would equal \( 2x \). Integrating \( 2x \) would lead to \( x^2 + C \), where \( C \) could be \( 3 \), \( 5 \), or any number. Thus, this ensures all possible functions that share the same derivative are represented. It is a crucial concept that emphasizes the generality of indefinite integrals.