Problem 40 - Calculus for Biology and Medicine

Question

In Problems \(37-42, a, b\), and \(c\) are constants and \(g(x)\) is a continuous function whose derivative \(g^{\prime}(x)\) is also continuous. Use substitution to evaluate each indefinite integral. $$ \int g^{\prime}(x) \cos [g(x)] d x $$

Step-by-Step Solution

Answer

The integral is \( \sin(g(x)) + C \).

Step 1: Identify the substitution

We want to find a substitution that simplifies the integral. Notice the presence of both \(g'(x)\) and \(g(x)\) in our integral. A convenient substitution is \( u = g(x) \).

Step 2: Differentiate the substitution

Differentiate \( u = g(x) \) with respect to \( x \) to obtain \( du = g'(x) \, dx \). This allows us to replace the differential \( dx \) in our integral.

Step 3: Substitute into the integral

Replace \( g(x) \) with \( u \) and \( g'(x) \, dx \) with \( du \) in the integral, transforming the integral into \( \int \cos(u) \, du \).

Step 4: Integrate the new expression

Find the integral of the new expression: \( \int \cos(u) \, du = \sin(u) + C \), where \( C \) is the constant of integration.

Step 5: Substitute back in terms of \(x\)

Substitute \( u = g(x) \) back into the expression to get the result in terms of \( x \): \( \sin(g(x)) + C \).

Key Concepts

Substitution Method

The substitution method is a powerful tool in calculus for transforming an integral into a simpler form. It's akin to solving a complex puzzle by replacing a section with a simpler piece. Here's how you can think about it:

• Identify the inner function that complicates the integral, often a function nested within another, as seen with functions like composite functions.
• Introduce a new variable, say \( u \), to substitute the inner function, simplifying the integral's structure.
• Calculate the new differential, \( du \), by differentiating \( u = g(x) \), where \( g(x) \) is the function being substituted.

By effectively changing variables, the substitution method aligns the integral's structure with standard formulas, making it manageable to solve.

Indefinite Integral

The indefinite integral represents a family of functions and is expressed as the antiderivative of a given function. Unlike definite integrals, which compute a number, indefinite integrals include an integration constant, \( C \). This constant accounts for all possible vertical shifts of the antiderivative function, as integrating a function reverses differentiation up to a constant shift.

• The notation \( \int f(x) \, dx \) signifies the indefinite integral of \( f(x) \).
• The result of an indefinite integral includes a function plus \( C \), embodying all potential solutions.

An indefinite integral of a cosine function, as in our example converting \( \int \cos(u) \, du \) to \( \sin(u) + C \), is a typical outcome demonstrating integral calculus's ability to provide general solutions.

Chain Rule

The chain rule in calculus is essential when differentiating composite functions. If a function \( y \) is composed of two or more functions, the chain rule helps find its derivative efficiently. The chain rule states:\[ \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} \]Whenever a function is differentiated that involves another function inside it (like \( y = \sin(g(x)) \)), the chain rule dictates how to differentiate by multiplying the derivative of the outer function by the derivative of the inner function.

• The outer function derives the larger framework (e.g., sine in \( \sin(u) \)).
• The inner function differentiates within (\( g(x) \)), addressing changes in its distinct layer.

The chain rule is crucial during the substitution method for understanding how different parts of composite functions interact during differentiation and integration.