Chapter 2: Problem 6
Evaluate \(\int \cos ^{2} x d x\)
Short Answer
Expert verified
The result is \( \frac{x}{2} + \frac{1}{4} \sin(2x) + C \).
Step by step solution
01
Recognize the Trigonometric Identity
To solve the integral \( \int \cos^2 x \, dx \), we first need to recognize and use the trigonometric identity for \( \cos^2 x \). Recall that \( \cos^2 x = \frac{1 + \cos(2x)}{2} \). This identity helps simplify the integral.
02
Rewrite the Integral Using the Identity
Substitute the identity into the integral. This gives us: \[ \int \cos^2 x \, dx = \int \frac{1 + \cos(2x)}{2} \, dx. \] This expression is simpler to integrate.
03
Separate the Integral
Break the integral into two separate integrals for easier computation. This results in: \[ \int \frac{1}{2} \, dx + \int \frac{\cos(2x)}{2} \, dx. \]
04
Integrate Each Part
Integrate each part separately. The first integral \( \int \frac{1}{2} \, dx \) gives \( \frac{x}{2} \), and the second integral \( \int \frac{\cos(2x)}{2} \, dx \) requires a substitution method. Let \( u = 2x \), therefore \( du = 2 \, dx \) or \( dx = \frac{du}{2} \). Thus, the integral becomes \( \int \frac{\cos u}{4} \, du \), which results in \( \frac{1}{4} \sin u + C_1 \). Substitute back \( u = 2x \) to get \( \frac{1}{4} \sin(2x) \).
05
Combine the Results
Combine the results from both integrals: \[ \frac{x}{2} + \frac{1}{4} \sin(2x) + C. \] Here, \( C \) is the constant of integration.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Trigonometric Identities
Trigonometric identities are equations that hold true for all values of the involved angles, serving as powerful tools in simplifying various mathematical expressions. In calculus, they are particularly useful in transforming integrals that involve trigonometric functions into forms that are easier to evaluate.
For example, when we see \( \cos^2 x \, dx \), we might feel stuck because directly integrating \( \cos^2 x \) isn't straightforward. This is where the trigonometric identity for \( \cos^2 x \) becomes handy: \( \cos^2 x = \frac{1 + \cos(2x)}{2} \).
Using this identity, we can rewrite the integral as \( \int \frac{1 + \cos(2x)}{2} \, dx \). This approach turns a complex integral into an easier task by splitting the expression into manageable parts, thanks to the identity.
For example, when we see \( \cos^2 x \, dx \), we might feel stuck because directly integrating \( \cos^2 x \) isn't straightforward. This is where the trigonometric identity for \( \cos^2 x \) becomes handy: \( \cos^2 x = \frac{1 + \cos(2x)}{2} \).
Using this identity, we can rewrite the integral as \( \int \frac{1 + \cos(2x)}{2} \, dx \). This approach turns a complex integral into an easier task by splitting the expression into manageable parts, thanks to the identity.
Substitution Method
The substitution method in integration is a technique similar to applying the chain rule in reverse for differentiation. It helps in simplifying integrals by transforming a complicated expression into a simpler one.
When integrating expressions such as \( \int \frac{\cos(2x)}{2} \, dx \), substitution becomes particularly useful. In this example, we set \( u = 2x \), then determine \( du = 2 \, dx \) or equivalently \( dx = \frac{du}{2} \).
Substituting for \( u \), the integral becomes \( \int \frac{\cos u}{4} \, du \), which is much simpler. We then integrate \( \frac{\cos u}{4} \, du \) to get \( \frac{1}{4} \sin u + C_1\). Lastly, we substitute back \( u = 2x \) to conclude with \( \frac{1}{4} \sin(2x)\).
This technique helps us solve integrals that initially appear complex by re-parameterizing them in terms of simpler variables.
When integrating expressions such as \( \int \frac{\cos(2x)}{2} \, dx \), substitution becomes particularly useful. In this example, we set \( u = 2x \), then determine \( du = 2 \, dx \) or equivalently \( dx = \frac{du}{2} \).
Substituting for \( u \), the integral becomes \( \int \frac{\cos u}{4} \, du \), which is much simpler. We then integrate \( \frac{\cos u}{4} \, du \) to get \( \frac{1}{4} \sin u + C_1\). Lastly, we substitute back \( u = 2x \) to conclude with \( \frac{1}{4} \sin(2x)\).
This technique helps us solve integrals that initially appear complex by re-parameterizing them in terms of simpler variables.
Definite Integration
Although this exercise discusses indefinite integrals, understanding how definite integration relates is essential. Definite integration calculates the accumulated quantity over an interval, contributing significantly to solving real-world problems like finding areas under curves.
In general, if we wanted to evaluate a definite integral similar to \( \int_a^b \cos^2 x \, dx \), we would first apply the same identity and substitution techniques.
However, after finding the antiderivative, we would then apply the limits of integration \( a \) and \( b \). This involves computing the difference \( F(b) - F(a) \), where \( F(x) \) is the antiderivative obtained after integration.
This approach provides the net area under the curve between the limits. Understanding this concept enhances problem-solving skills and helps apply mathematical techniques to real situations.
In general, if we wanted to evaluate a definite integral similar to \( \int_a^b \cos^2 x \, dx \), we would first apply the same identity and substitution techniques.
However, after finding the antiderivative, we would then apply the limits of integration \( a \) and \( b \). This involves computing the difference \( F(b) - F(a) \), where \( F(x) \) is the antiderivative obtained after integration.
This approach provides the net area under the curve between the limits. Understanding this concept enhances problem-solving skills and helps apply mathematical techniques to real situations.
Trigonometry in Calculus
In calculus, trigonometric functions often surface within limits, derivatives, and integrals, showcasing the interconnectedness between trigonometry and calculus. Trigonometry in calculus provides insightful ways to explore periodic behaviors, oscillations, and waves.
In our exercise, understanding how trigonometric identities simplify integrals is a perfect example of this relationship. \( \cos^2 x\), a trigonometric expression, becomes manageable through identities and calculus principles, paving the way for deeper analysis.
Trigonometric integrals allow us to explore a plethora of natural phenomena because many periodic processes can be described using sine and cosine functions. These integrals help in solving differential equations, modeling circular motion, and analyzing signals.
Ultimately, appreciating trigonometry in calculus enables students to tackle a broader range of mathematical challenges with confidence and insight, underscoring the unifying theme of mathematics.
In our exercise, understanding how trigonometric identities simplify integrals is a perfect example of this relationship. \( \cos^2 x\), a trigonometric expression, becomes manageable through identities and calculus principles, paving the way for deeper analysis.
Trigonometric integrals allow us to explore a plethora of natural phenomena because many periodic processes can be described using sine and cosine functions. These integrals help in solving differential equations, modeling circular motion, and analyzing signals.
Ultimately, appreciating trigonometry in calculus enables students to tackle a broader range of mathematical challenges with confidence and insight, underscoring the unifying theme of mathematics.
