Problem 1 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

In Problems 1 through 18 , evaluate the integral. \(\int \cos ^{2} x d x\)

Step-by-Step Solution

Answer

\(\frac{x}{2} + \frac{\sin(2x)}{4} + C\)

Step 1: Substitute \(\cos^{2}(x)\) with \(\frac{1+\cos(2x)}{2}\)

Using the trigonometric power reduction formula, we rewrite \(\cos^{2}(x)\) as \(\frac{1+\cos(2x)}{2}\). So the integral becomes \(\int \frac{1+\cos(2x)}{2} dx \)

Step 2: Apply the integral of sum rule

The integral of a sum is equal to the sum of integrals. Here, we separate the integral into two parts: \(\int \frac{1}{2} dx + \int \frac{\cos(2x)}{2} dx \)

Step 3: Evaluate each integral separately

Evaluate each of the two integrals separately. The integral of \(\frac{1}{2}\) is \(\frac{x}{2}\) and for integral of \(\frac{\cos(2x)}{2}\), use u-substitution, let \(u = 2x\) gives \(dx = \frac{du}{2}\). Substituting these into the integral gives \(\int \frac{\cos(u)}{4} du\), which leads to \(\frac{\sin(u)}{4}\), substituting back \(u = 2x\) we get \(\frac{\sin(2x)}{4}\)

Step 4: Combine the solutions and add the constant of integration C

Finally, combine the solutions of the two integrals and add the constant of integration \(C\) to the result in order to form the most general solution possible. The final result is \(\frac{x}{2} + \frac{\sin(2x)}{4} + C\)

Key Concepts

Integration Techniques

Integration techniques are essential mathematical tools that allow us to find the area under curves described by functions. In calculus, integrals can sometimes appear complex, but with the right approach, they become manageable.

To integrate certain functions, we often break them down using various rules or techniques. For example, the formula for the sum of integrals states:

• The integral of a sum is the sum of the integrals: \[ \\int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx \]

This allows us to handle each part of the integral separately, simplifying the process immensely. Breaking down the integral into manageable chunks can turn a complex problem into a series of simpler ones. By using techniques like the sum rule or substitution, tackling integrals becomes a straightforward task.

Trigonometric Identities

Trigonometric identities serve as handy tools in calculus, especially when dealing with integrals involving trigonometric functions. In our current problem, we dealt with \( \cos^{2}(x) \), which can be rewritten using a trigonometric identity known as the power reduction formula:

• \[ \\cos^{2}(x) = \frac{1 + \cos(2x)}{2} \]

This formula simplifies our integral by transforming it into a form that can be easily integrated. Without using this identity, the integral would be much harder to solve. Identifying and applying relevant identities can vastly simplify your calculations, particularly in trigonometric contexts.

Mastering these identities is crucial, as they enable the transformation of complex trigonometric expressions into simpler forms, ready for integration or differentiation.

U-Substitution

U-substitution is a valuable method for solving integrals, particularly useful when dealing with composite functions. This technique is akin to the chain rule in differentiation but applied in reverse.

Consider the integral involving \( \cos(2x) \). To simplify this, we use u-substitution:

• Let \( u = 2x \). Then, \( du = 2 \, dx \) or \( dx = \frac{du}{2} \).
• This transformation changes the integral: \[ \\int \frac{\cos(2x)}{2} \, dx = \int \frac{\cos(u)}{4} \, du \]

Now, integrating \( \cos(u) \) becomes much simpler, resulting in \( \frac{\sin(u)}{4} \). After integrating, substitute back \( u = 2x \) to obtain the solution in terms of the original variable.

U-substitution simplifies integration by reducing it to a basic form, helping to easily address otherwise complex integral problems.