Problem 11 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-28, perform the indicated integrations. \(\int \sin ^{4} 3 t \cos ^{4} 3 t d t\)

Step-by-Step Solution

Answer

\( \frac{5t}{256} - \frac{\sin(12t)}{384} + \frac{\sin(24t)}{3072} + C \)

Step 1: Simplify Using Trigonometric Identity

To integrate \( \int \sin^4(3t) \cos^4(3t) \, dt \), we first simplify the expression using the trigonometric identity: \( \sin^2(x) = \frac{1 - \cos(2x)}{2} \) and \( \cos^2(x) = \frac{1 + \cos(2x)}{2} \). Therefore, \( \sin^4(3t) = \left( \frac{1 - \cos(6t)}{2} \right)^2 \) and \( \cos^4(3t) = \left( \frac{1 + \cos(6t)}{2} \right)^2 \).

Step 2: Expand the Expression

Multiply the expanded expressions: \( \sin^4(3t) \cos^4(3t) = \left( \frac{1 - \cos(6t)}{2} \right)^2 \left( \frac{1 + \cos(6t)}{2} \right)^2 \). This simplifies to \( \left( \frac{(1 - \cos^2(6t))(1 + \cos^2(6t))}{16} \right) = \frac{(1 - \cos^2(6t))^2}{16} \).

Step 3: Apply Power Reducing Identity

Use the identity \( \cos^2(x) = \frac{1 + \cos(2x)}{2} \) to replace \( \cos^2(6t) \):\( (1 - \cos^2(6t))^2 = (\sin^2(6t))^2 \). Now integrate \( \int \frac{(\sin^2(6t))^2}{16} \, dt \).

Step 4: Substitute and Integrate

Change \( \sin^2(6t) \) using the identity \( \sin^2(x) = \frac{1 - \cos(2x)}{2} \), so \( \sin^2(6t) = \frac{1 - \cos(12t)}{2} \). Thus, the integral becomes \( \int \frac{(\frac{1 - \cos(12t)}{2})^2}{16} \, dt = \int \frac{1 - 2\cos(12t) + \cos^2(12t)}{64} \, dt \).

Step 5: Break Down the Integral

Separate the integral: \( \int \frac{1}{64} - \frac{2\cos(12t)}{64} + \frac{\cos^2(12t)}{64} \, dt \). Integrate each part individually: \( \int \frac{1}{64} \, dt \); \( - \int \frac{2\cos(12t)}{64} \, dt \); \( \int \frac{\cos^2(12t)}{64} \, dt \).

Step 6: Solve Each Integral

For \( \int \frac{1}{64} \, dt \), the result is \( \frac{t}{64} + C_1 \). For \( -\int \frac{2\cos(12t)}{64} \, dt \), use the substitution \( u = 12t \), then integrate to get \(- \frac{1}{32} \left( \frac{\sin(12t)}{12} \right) \). For \( \int \frac{\cos^2(12t)}{64} \, dt \), use the power-reducing identity \( \int \frac{1 + \cos(24t)}{2 \cdot 64} \, dt \), yielding \( \frac{1}{128}(t + \frac{\sin(24t)}{24}) \).

Step 7: Combine Results and Simplify

Combine the results from Step 6: \( \frac{t}{64} - \frac{\sin(12t)}{384} + \frac{t}{256} + \frac{\sin(24t)}{3072} + C \). Simplify to \( \frac{5t}{256} - \frac{\sin(12t)}{384} + \frac{\sin(24t)}{3072} + C \).

Step 8: Conclusion

Thus, the integral \( \int \sin^4(3t) \cos^4(3t) \, dt \) simplifies to the final form: \( \frac{5t}{256} - \frac{\sin(12t)}{384} + \frac{\sin(24t)}{3072} + C \).

Key Concepts

Trigonometric Identities

Trigonometric identities are crucial tools that help simplify complex trigonometric expressions. They provide relationships between sine, cosine, and other trigonometric functions.
For instance, the identities \(\sin^2(x) = \frac{1 - \cos(2x)}{2}\) and \(\cos^2(x) = \frac{1 + \cos(2x)}{2}\) are used extensively to transform expressions into more manageable forms.
These particular identities help to reduce the powers of sine and cosine, allowing for easier integration by transforming products of powers into polynomials in terms of cosine, which are easier to integrate.
Using these identities, one can break down expressions involving products like \(\sin^4(x)\) and \(\cos^4(x)\) into simpler components. Understanding these identities is essential for tackling integration problems involving trigonometric functions.

Power Reducing Identities

Power reducing identities are specific trigonometric identities that aid in rewriting expressions with higher powers of trigonometric functions to simpler, lower power versions.
This is particularly helpful in integration, where it's easier to evaluate integrals of simpler expressions.
For example, consider the power reducing identity for cosine:\(\cos^2(x) = \frac{1 + \cos(2x)}{2}\). By using this, a higher power like \(\cos^4(x)\) can be expressed as \((\frac{1 + \cos(2x)}{2})^2\).
This collapses the complexity drastically, turning the problem into evaluating simpler functions.
Therefore, power reducing identities simplify the process of integration by transforming challenging integrals into expressions involving trigonometric identities with reduced power, making them more straightforward to handle.

Substitution Method

The substitution method, also known as u-substitution, is a powerful technique in calculus used to evaluate integrals, especially when direct integration is challenging.
This method involves substituting a part of the integral with a new variable, simplifying the integration process.
In our problem, when integrating expressions like \(-\int \frac{2\cos(12t)}{64} \, dt\), substitution is often necessary. By letting \(u = 12t\), the differentials adjust accordingly, simplifying the integral to a form that is easier to integrate.
The substitution method relies on identifying an inner function whose derivative is also present in the integrand, thereby making it easier to integrate. It's a vital skill for handling complex integrals, especially those in trigonometric contexts.

Definite and Indefinite Integrals

Integrals can be classified into two types: definite and indefinite integrals.
Indefinite integrals represent a family of functions and are expressed as \(\int f(x) \, dx = F(x) + C\), where \(C\) is the constant of integration. They provide the anti-derivative of a function.

• In our context, the integral \(\int \sin^4(3t) \cos^4(3t) \, dt\) is indefinite since it's expressed with a constant \(C\).

Definite integrals, on the other hand, calculate the area under the curve of a function for a specified interval \([a, b]\). This is particularly useful for finding accumulated quantities or areas in physics and engineering.
While the current problem involves indefinite integrals, understanding both types helps in comprehending how integrals work to describe antiderivatives and areas.