Proving and Evaluating the Integral of sin(ax)cos(ax)

Understanding how to manipulate and integrate trigonometric functions is a fundamental skill in calculus. One particular challenge is proving and evaluating the integral of the product of sin(ax) cos(ax). This article will guide you through the process, emphasizing the steps and techniques used to simplify and integrate trigonometric expressions effectively.

Introduction to the Integral of sin(ax)cos(ax)

Consider the function sin(ax) cos(ax). At first glance, it might seem that finding its integral would be cumbersome. However, by leveraging trigonometric identities and the concept of antiderivatives, we can unravel the complexity of this integral.

Step 1: Differentiating sin^2(ax)

Let's start by differentiating sin^2(ax). Using the chain rule, we have:

[ frac{d}{dx}(sin^2(ax)) 2a sin(ax) cos(ax) ]

This result tells us that 1/2a sin^2(ax) cos(ax) is an antiderivative of sin(ax) cos(ax). This is our first step in exploring the integral.

Step 2: Simplifying the Expression

Given that 2 sin(x) cos(x) sin(2x), we can rewrite sin(ax) cos(ax) as:

[ frac{sin(2ax)}{2} ]

Therefore, the integral we need to evaluate becomes:

[ I frac{1}{2} int sin(2ax) , dx ]

Now, using the substitution t 2ax, we get:

[ I frac{1}{2} times frac{1}{2a} left[-cos(2ax)right] C ]

Step 3: Alternative Simplification Using Trigonometric Identity

Recall the double angle identity for cosine: cos(2x) 1 - 2 sin^2(x). Applying this to cos(2ax), we have:

[ cos(2ax) 1 - 2 sin^2(ax) ]

Thus, the integral can be rewritten as:

[ I frac{1}{2} times frac{1}{2a} left[-(1 - 2 sin^2(ax))right] C ]

Further simplifying, we get:

[ I frac{1}{2a} sin^2(ax) - frac{1}{4a} C ]

This form is equivalent to our earlier result, showing the consistency of our steps.

Conclusion and Practical Applications

The process of integrating sin(ax)cos(ax) involves recognizing trigonometric identities and applying the rules of integration. This technique is not only crucial for solving specific problems but also serves as a foundational skill in more complex calculus problems.

Tips for Mastering Trigonometric Integrals

Memorize Key Identities: Familiarity with fundamental trigonometric identities (like 2 sin(x) cos(x) sin(2x) and cos(2x) 1 - 2 sin^2(x)) can significantly speed up the solving process. Substitution: Use substitution (such as t 2ax) to simplify integrals involving products of trigonometric functions. Precise Notation: Writing clear and structured steps, especially when dealing with chain rules and substitutions, ensures accuracy.

Understanding how to prove and evaluate the integral of sin(ax) cos(ax) is a valuable skill that enhances your ability to tackle more advanced calculus problems. By mastering the techniques outlined here, you can confidently handle a wide range of trigonometric integrals.