Problem 41 - 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. $$ \int g^{\prime}(x) e^{-g(x)} d x $$

Step-by-Step Solution

Answer

\(-e^{-g(x)} + C\)

Step 1: Understand the Problem

The integral \( \int g^{\prime}(x) e^{-g(x)} \, dx \) involves a composition of functions, specifically a function inside an exponential function. The derivative \( g^{\prime}(x) \) appears as a factor, suggesting a substitution method.

Step 2: Choose a Substitution

Let \( u = g(x) \). This makes the derivative \( du = g^{\prime}(x) \, dx \). This substitution simplifies the integral because \( g^{\prime}(x) \, dx \) can be directly replaced with \( du \).

Step 3: Change the Integral Variable

Substitute \( u = g(x) \) and \( du = g^{\prime}(x) \, dx \) into the integral. The integral \( \int g^{\prime}(x) e^{-g(x)} \, dx \) becomes \( \int e^{-u} \, du \).

Step 4: Integrate in Terms of \( u \)

Integrate \( \int e^{-u} \, du \), which results in \( -e^{-u} + C \), where \( C \) is the constant of integration.

Step 5: Substitute Back to Original Variable

Replace \( u \) with \( g(x) \) to express the result in terms of the original variable. The integral becomes \( -e^{-g(x)} + C \).

Key Concepts

Integration Techniques

Integration can sometimes be tricky, especially when dealing with complex functions or compositions. A powerful method to simplify the process is **substitution**, which transforms an integrand into a more manageable form.
Here's a step-by-step guide to using the substitution technique:

• **Identify the Inner Function:** Look for functions like polynomials or exponentials within the integral that have derivatives present. These are often candidates for substitution.
• **Choose the Substitution:** Set the inner function as a new variable, such as \( u = g(x) \). This helps in simplifying the function.
• **Differentiate Your Substitution:** Find the differential, like \( du = g^{\prime}(x) dx \), which should align with parts of your integral to ease substitution.
• **Transform the Integral:** Substitute both the function and its differential into your integral, changing all variables to \( u \) and \( du \).
• **Integrate:** Work out the integral in terms of the new variable.
• **Substitute Back:** Replace \( u \) with the original function to pose the antiderivative in terms of the original variable \( x \).

By practicing these steps, substitution can become an invaluable tool in your calculus toolkit.

Indefinite Integrals

Indefinite integrals are expressions representing a family of functions. They have the form: \( \int f(x) \, dx \). Unlike definite integrals, they lack specific upper and lower limits for the function. Instead, they're pivotal in identifying antiderivatives.
Key points about indefinite integrals include:

• **General Form:** The result of an indefinite integral \( \int f(x) \, dx \) is an antiderivative, written as \( F(x) + C \), where \( C \) is the constant of integration.
• **Freedom of Constants:** Since integration is the inverse of differentiation, adding any constant to an antiderivative doesn't affect its derivative, justifying the constant \( C \).
• **No Boundaries:** Unlike definite integrals, the integral of a function without bounds reflects the general curve family for the antiderivative.

Understanding indefinite integrals helps you grasp the continuous nature and infinite possibilities stemming from integration in calculus.

Continuous Functions

Continuous functions are those that are unbroken or uninterrupted across their domain, meaning they have no gaps, jumps, or breaks. This continuity is essential in calculus, where differentiability and integrability often presuppose continuity.
Here are some characteristics and implications of continuous functions:

• **Smoothness Assurance:** A continuous function means its graph can be drawn without lifting your pencil, providing smooth transitions.
• **Differentiability Ties:** If a function is continuous over its entire domain, it's often differentiable, meaning slopes or rates of change can be calculated at each point.
• **Consequence on Integration:** Integral calculus commonly assumes the integrability of continuous functions, allowing seamless calculation of areas under curves or other integral applications.

Overall, understanding the concept of continuous functions is pivotal for fixing concepts like substitution, integration, and more, which rely on smooth and predictable changes within functions.