91Ó°ÊÓ

The double-angle identity is a fundamental concept in trigonometry that simplifies expressions involving trigonometric functions. In this exercise, we specifically utilized the double-angle identity for sine:

• \[\sin 2x = 2\sin x\cos x\]

This formula allows us to express \( \sin 2x \) in terms of products of \( \sin x \) and \( \cos x \). It's very handy when simplifying integrals or solving trigonometric equations. In our exercise, it was the key to transforming the original expression \( \frac{\sin 2x}{\sin x} \) into \( 2\cos x \). By performing this simplification first, we made the integration process much easier. Knowing when and how to use such identities can turn complex-looking problems into straightforward tasks.

Trigonometric integrals are integrals that involve trigonometric functions. They appear frequently in calculus, often requiring different techniques for solving. In our problem, after using the double-angle identity, the integral transforms into one that is easier to handle:

• \[\int 2 \cos x \, dx\]

Trigonometric integrals can vary from very simple to quite complex. In our case, the presence of \( \cos x \) makes the integral straightforward because we know that the integral of \( \cos x \) is \( \sin x \). Understanding the common integrals of trigonometric functions, such as \( \int \sin x \, dx = -\cos x + C\) and \( \int \cos x \, dx = \sin x + C\), is crucial. This knowledge helps one to seamlessly integrate functions like \( \cos x \) without the need for more complicated techniques.

In calculus, various integration techniques are used to find indefinite integrals. These techniques include substitution, integration by parts, partial fraction decomposition, and more. In this exercise, we kept it simple by using the recognition of standard integrals. Standard Integrals: Many fundamental functions have been integrated numerous times, allowing us to remember their indefinite integrals as formulas. For example, recognizing that \( \int \cos x \, dx = \sin x + C \), lets us quickly find the integral of \( 2 \cos x \) by simple multiplication with 2:

• \[2 \int \cos x \, dx = 2\sin x + C\]

Content knowledge, like recalling these formulas, streamlines the process of finding solutions.Simplification: Before jumping into complex techniques, simplifying the integrand, as we did using the double-angle identity, can often make the integration process much more accessible. Simplification allowed us to turn the original problem into a much easier integral to evaluate.