Problem 24 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(13-48\) . $$ \int \sin (8 z-5) d z $$

Step-by-Step Solution

Answer

The integral is \(-\frac{1}{8} \cos(8z - 5) + C\).

Step 1: Identify the Integration Method

We need to integrate the function \( \sin(8z - 5) \) with respect to \( z \). The standard method for integrating trigonometric functions like sine is to use substitution if the argument is more complex than just a single variable.

Step 2: Perform a Substitution

Let's use substitution to simplify the integration. Let \( u = 8z - 5 \). As a result, the differential \( du = 8 \, dz \) or \( dz = \frac{1}{8} du \).

Step 3: Substitute and Simplify

Substituting \( u \) and \( dz \) in the integral gives us:\[ \int \sin(u) \cdot \frac{1}{8} \, du \]This simplifies to:\[ \frac{1}{8} \int \sin(u) \, du \]

Step 4: Integrate the Simplified Function

The integral of \( \sin(u) \) is \( -\cos(u) \). Therefore,\[ \frac{1}{8} \int \sin(u) \, du = -\frac{1}{8} \cos(u) + C \]where \( C \) is the constant of integration.

Step 5: Substitute Back the Original Variable

Re-substitute \( u = 8z - 5 \) back into the integrated function:\[ -\frac{1}{8} \cos(8z - 5) + C \]This gives us the final result after substitution back.

Key Concepts

Trigonometric Integration

Trigonometric integration is a technique used to integrate functions that involve trigonometric identities like sine and cosine. When dealing with these functions, especially if they involve more than just the elementary form (like \( \sin(ax+b) \)), it's often necessary to simplify the expression first.

• The goal is to convert the integral into an easier form.
• This might involve using identities or applying substitution methods.

Trigonometric integrals are encountered frequently in calculus and have applications in physics and engineering.
The trigonometric function in this exercise, \( \sin(8z - 5) \), required simplification through substitution, a common tactic for handling such integrals.

U-Substitution

U-Substitution is a method used for simplifying integrals, particularly when the integral involves a composite function. The substitution simplifies the integrand into a basic form that is easier to integrate.
This method involves:

• Identifying a suitable substitution (e.g., let \( u = 8z - 5 \) in this exercise).
• Determining the differential \( du \) from \( dz \).
• Replacing all instances of the original variable with \( u \).
• Integrating with respect to \( u \).

This conversion often transforms a complex trigonometric argument into something much simpler. After integration, we substitute back the original variable to express the answer in its original context.

Integration of Sine Function

The integration of the sine function is a fundamental operation in calculus. The integral of \( \sin(u) \) is \( -\cos(u) + C \). It's important to note this result because it is frequently used across different calculations.
The particularity of sine integration is that it produces a negative cosine due to the derivative relationship,

• The derivative of \( \cos(u) \) is \( -\sin(u) \).
• Therefore, integrating \( \sin(u) \) reverses this process, yielding \( -\cos(u) \).
• This outcome holds regardless of the complexity of \( u \).

By identifying the integral correctly, we can easily complete the remaining operations of our integration.

Constant of Integration

In indefinite integrals, the constant of integration is a key component as it represents an entire family of solutions. This constant, denoted by \( C \), recognizes that there are many functions that have the same derivative.
It ensures we consider all possible antiderivatives:

• Without \( C \), the integral would suggest a single solution.
• Including \( C \) maintains the general form of all possible solutions.

In our exercise, \( -\frac{1}{8} \cos(8z - 5) + C \), \( C \) abstracts the infinite vertical translations of the cosine function graph.