Problem 39 - Calculus for Biology and Medicine

Question

a, b, and \(c\) are constants and \(g(x)\) is a continuous function whose derivative \(g^{\prime}(x)\) is also continuous. Use substitution to evaluate the indefinite integrals. $$ \int g^{\prime}(x)[g(x)]^{n} d x $$

Step-by-Step Solution

Answer

\( \int g'(x) [g(x)]^n \, dx = \frac{[g(x)]^{n+1}}{n+1} + C \), where \( C \) is a constant.

Step 1: Recognize the structure

The given integral is \( \int g'(x) [g(x)]^n \, dx \). Notice that this can fit the form \( \int u^n \, du \) if we correctly identify \( u \).

Step 2: Define substitution variables

Let \( u = g(x) \). Consequently, the derivative \( du = g'(x) \, dx \), which aligns with the term \( g'(x) \, dx \) in the integral.

Step 3: Rewrite the integral using substitution

Substitute \( u = g(x) \) and \( du = g'(x) \, dx \) into the integral, transforming it into \( \int u^n \, du \).

Step 4: Integrate using the power rule

Apply the power rule for integration to \( \int u^n \, du \), which is \( \frac{u^{n+1}}{n+1} + C \), where \( C \) is the integration constant.

Step 5: Substitute back to the original variable

Replace \( u \) with \( g(x) \) in the expression, giving the final result as \( \frac{[g(x)]^{n+1}}{n+1} + C \).

Key Concepts

Indefinite Integrals

Indefinite integrals represent one of the key concepts in calculus, and they are often referred to as antiderivatives. When you solve an indefinite integral, it means you are trying to find a function that, when differentiated, will give you the original function back. This differs from definite integrals, which have limits of integration and yield a numerical value. However, with indefinite integrals, because no limits are specified, the result is a family of functions rather than a single number. The general form of an indefinite integral is written as \( \int f(x) \, dx \), where \( f(x) \) is the function you're integrating. The result of the integral includes an integration constant \( C \), because differentiation of any constant is zero. This constant accounts for all potential vertical shifts of the antiderivative in the graph. Indefinite integrals are essential in reversing the process of differentiation, providing a tool for determining the original function before its rate of change was calculated.

Substitution Method

The substitution method is a powerful technique used to simplify the process of finding indefinite integrals. It is especially useful when dealing with integrals that aren't straightforward and require simplification. The key idea behind substitution is to replace a part of the given function with a new variable to make the integral easier to evaluate.Here's a step-by-step breakdown of the substitution method:

• Identify a part of the function to replace with a new variable, typically one that simplifies the integral. Let's call this part \( u \).
• Find the differential \( du \) by differentiating \( u \) with respect to \( x \). This step is crucial for ensuring the substitution accounts accurately for the entire differential.
• Rewrite the original integral in terms of \( u \) and \( du \). This usually simplifies the integral to a form that is easier to compute.
• Perform the integration in terms of \( u \), and then substitute back the original variable from the function to express the final answer in its original terms.

These steps help to simplify complex integrals into basic forms that are much easier to handle, especially through the lens of integration rules.

Integration by Substitution

Integration by substitution is specifically applying the substitution method as a technique for solving integrals, transforming the given integral into one that is more manageable. In the original problem, \( \int g'(x) [g(x)]^n \, dx \), substitution offers a way to streamline the process by recognizing a suitable substitution.To use integration by substitution here:

• We define \( u = g(x) \). This choice is critical as it aligns with the goal to simplify the form of the integral.
• We derive that \( du = g'(x) \, dx \). This step confirms that the differential part of the integral matches exactly \( du \), thereby covering all parts of the expression correctly.
• Once substituted, the integral becomes \( \int u^n \, du \). This is a straightforward power rule integration.
• The solution of \( \int u^n \, du \) is \( \frac{u^{n+1}}{n+1} + C \), reflecting the antiderivative of the new expression.
• Finally, switch back from \( u \) to the original function \( g(x) \), yielding the result \( \frac{[g(x)]^{n+1}}{n+1} + C \).

This method ensures accuracy through simplification and helps in managing integrals that are not easily approachable through basic rules alone.