Question

Use the identities \(\sin ^{2} x=(1-\cos 2 x) / 2\) and \(\cos ^{2} x=(1+\cos 2 x) / 2\) to find \(\int \sin ^{2} x d x\) and \(\int \cos ^{2} x d x\)

Short answer

Question: Find the integrals of \(\sin^2 x\) and \(\cos^2 x\) using the identities \(\sin^2 x = \frac{1-\cos 2x}{2}\) and \(\cos^2 x = \frac{1+\cos 2x}{2}\). Answer: 1. \(\int \sin^2 x dx = \frac{1}{2} x - \frac{1}{4} \sin 2x + C\) 2. \(\int \cos^2 x dx = \frac{1}{2} x + \frac{1}{4} \sin 2x + C\)

Step-by-step solution

Substitute the identities

Replace \(\sin^2 x\) with \(\frac{1-\cos 2x}{2}\) and \(\cos^2 x\) with \(\frac{1+\cos 2x}{2}\). We now have to find the integrals: 1. \(\int \sin^2 x dx = \int \frac{1-\cos 2x}{2} dx\) 2. \(\int \cos^2 x dx = \int \frac{1+\cos 2x}{2} dx\)

Separate the terms in both integrals

Separate the terms for both integrals to make the integration process easier. 1. \(\int \sin^2 x dx = \int \frac{1}{2} dx - \int \frac{\cos 2x}{2} dx\) 2. \(\int \cos^2 x dx = \int \frac{1}{2} dx + \int \frac{\cos 2x}{2} dx\)

Integrate each term

Perform integration for each term in the separated integrals. 1. \(\int \sin^2 x dx = \int \frac{1}{2} dx - \int \frac{\cos 2x}{2} dx = \frac{1}{2} x - \frac{1}{4} \sin 2x + C\) 2. \(\int \cos^2 x dx = \int \frac{1}{2} dx + \int \frac{\cos 2x}{2} dx = \frac{1}{2} x + \frac{1}{4} \sin 2x + C\) Here, \(C\) is the integration constant. Now, we have found the integrals: 1. \(\int \sin^2 x dx = \frac{1}{2} x - \frac{1}{4} \sin 2x + C\) 2. \(\int \cos^2 x dx = \frac{1}{2} x + \frac{1}{4} \sin 2x + C\)

Key concepts

Integration Techniques

Integration techniques are fundamental methods in calculus used to find antiderivatives or integrals of functions. They transform complex integrals into simpler forms that are easier to compute.
For trigonometric integrals, like integrating \(\sin^2 x\) or \(\cos^2 x\), common strategies involve using substitution, trigonometric identities, or parts of integration.

In our example, we employed a strategic approach involving substitution using trigonometric identities. This helped to simplify the functions.

• Firstly, identities like \(\sin^2 x = \frac{1-\cos 2x}{2}\) are used to rewrite the original integral in a simpler form.
• Then, separating each term within the rewritten integral simplifies further computation.
• Finally, integrating term by term using basic integral rules simplifies the entire process.

This approach can be widely applied in tackling a variety of trigonometric integrals.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the input angle. These identities are extremely useful in simplifying trigonometric expressions, allowing easier manipulation and computation.
For instance, in the integration problem, we used the following identities:

• \(\sin^2 x = \frac{1-\cos 2x}{2}\)
• \(\cos^2 x = \frac{1+\cos 2x}{2}\)

These identities help reduce the complexity by transforming products of trigonometric functions into linear combinations.
Without these identities, integrating squared trigonometric functions directly would be much more complicated. Knowing and applying these formulas is a great asset in solving a wide range of problems efficiently.

Calculus

Calculus is the branch of mathematics that deals with the study of change and motion, primarily through two types of operations: differentiation and integration.
It provides the tools for calculating the rate of change and the accumulation of quantities.
In our example, calculus techniques were applied through integration to solve integrals involving squared trigonometric functions.
This involves the process of finding the integral of functions, offering solutions to problems such as area under curves or cumulative total of changing quantities.

• The antiderivative process was showcased by finding the integral of \(\sin^2 x\) and \(\cos^2 x\), which entails calculating the indefinite integral of trigonometric expressions.
• This leads to expressions like \(\frac{1}{2} x - \frac{1}{4} \sin 2x + C\) and \(\frac{1}{2} x + \frac{1}{4} \sin 2x + C\), respectively.
• The constant \(C\) signifies an indefinite integral, representing any constant value that could be added to the function.

These calculations play a crucial role in many scientific and engineering applications, providing solutions to real-world dynamics and continuous growth problems.