Question

Evaluate the given integral. $$ \int \cos ^{4} 2 x d x $$

Short answer

\( \frac{3}{8}x + \frac{1}{8}\sin(4x) + \frac{1}{64}\sin(8x) + C \)

Step-by-step solution

Simplify the Power Using a Trigonometric Identity

The first step is to simplify \( \cos^4(2x) \) using a power-reduction identity. We know that \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \). Therefore, \(\cos^4(2x) = (\cos^2(2x))^2 = \left(\frac{1 + \cos(4x)}{2}\right)^2.\)

Expand the Square

Use algebraic expansion to simplify \( \left(\frac{1 + \cos(4x)}{2}\right)^2 \). This gives:\[\left(\frac{1 + \cos(4x)}{2}\right)^2 = \frac{1}{4}(1 + 2\cos(4x) + \cos^2(4x)).\]

Simplify Further Using Identities

Next, simplify \( \cos^2(4x) \) using the identity again.\(\cos^2(4x) = \frac{1 + \cos(8x)}{2},\)so the expression becomes:\[\frac{1}{4}(1 + 2\cos(4x) + \frac{1 + \cos(8x)}{2}).\]

Expression Expansion

This expands to:\[\frac{1}{4}(1 + 2\cos(4x) + \frac{1}{2} + \frac{1}{2}\cos(8x)) = \frac{3}{8} + \frac{1}{2}\cos(4x) + \frac{1}{8}\cos(8x).\]

Integrate Term by Term

We now integrate each term separately: - \( \int \frac{3}{8} \, dx = \frac{3}{8}x + C_1 \).- \( \int \frac{1}{2}\cos(4x) \, dx = \frac{1}{2} \cdot \frac{1}{4} \sin(4x) + C_2 = \frac{1}{8}\sin(4x) + C_2 \).- \( \int \frac{1}{8}\cos(8x) \, dx = \frac{1}{8} \cdot \frac{1}{8} \sin(8x) + C_3 = \frac{1}{64}\sin(8x) + C_3 \).Combine these results to form the entire integral.

Formulate the Final Result

Combine the integrated terms into a single final expression:\[\int \cos^4(2x)\, dx = \frac{3}{8}x + \frac{1}{8}\sin(4x) + \frac{1}{64}\sin(8x) + C,\]where \( C \) is the constant of integration.

Key concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in calculus, particularly when dealing with integrals involving trigonometric functions. These identities allow us to simplify complex expressions and make integration more manageable.
For example, the identity \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \) plays a crucial role in simplifying expressions involving higher powers of cosine. This identity is often used to reduce the power of cosine functions, transforming them into a format that is easier to integrate.
An important use of trigonometric identities is in transforming products of trigonometric functions into sums, called product-to-sum formulas. These transformations often lead to simpler integrals. Understanding and applying these identities can significantly ease the integration process.

Power-Reduction Identity

The power-reduction identity is a specific type of trigonometric identity used to simplify the integration of powers of sine and cosine functions. These identities express higher powers of trigonometric functions in terms of lower powers, making them easier to integrate. For instance, the identity \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \) reduces the power of the cosine function from two to one.
In the given exercise, this technique was used to simplify \( \cos^4(2x) \) into an expression involving \( \cos(4x) \) and \( \cos(8x) \). Step by step, this simplification converts the integrand into a sum of cosines of different multiples, greatly simplifying the integration later on.
Using power-reduction identities helps in breaking down complex trigonometric functions into simpler components that can be integrated term by term.

Integration Techniques

Integration techniques are methods used to evaluate integrals. These techniques are crucial for solving integrals that are not immediately recognizable.
Some common techniques include:

• Substitution: Replacing a complex part of the integrand with a single variable.
• Integration by Parts: Breaking down the product of functions using the formula \( \int u \ dv = uv - \int v \ du \).
• Trigonometric Identities: Using identities to simplify integrands involving trigonometric functions, as seen in the given exercise.
  

In this exercise, after using trigonometric identities, the integral became the sum of simpler integrals that were solved separately. Each part was integrated using basic integration rules: constant integration and the integration of cosine functions.
Being familiar with a variety of integration techniques allows one to approach different integrals with confidence and effectively simplify the process.

Definite and Indefinite Integrals

Understanding the difference between definite and indefinite integrals is key in integral calculus. An indefinite integral represents a family of functions and includes a constant of integration \( C \). It is expressed as:\[ \int f(x) \, dx = F(x) + C \]where \( F(x) \) is the antiderivative of \( f(x) \).
A definite integral, on the other hand, computes the area under the curve of \( f(x) \) between two specific points \( a \) and \( b \). It is expressed as:\[ \int_{a}^{b} f(x) \, dx \]resulting in a number that represents the accumulated value.
In our exercise, we dealt with an indefinite integral since no limits were specified. Thus, the final answer included a constant \( C \), representing the family of all possible antiderivatives. Understanding when to use definite versus indefinite integrals depends on the problem's requirements and what is being calculated.