Question

Perform the indicated integrations. $$ \int \sin ^{4} 3 t \cos ^{4} 3 t d t $$

Short answer

The solution is \( \frac{t}{32} - \frac{1}{192} \sin(12t) + \frac{1}{3072} \sin(24t) + C \).

Step-by-step solution

Recognize the Integral Type

Notice that the integral involves powers of sine and cosine with the same argument. This calls for the use of trigonometric identities to simplify the integration process.

Use the Double Angle Identity

Use the identity \( \sin^2 x = \frac{1 - \cos(2x)}{2} \) and \( \cos^2 x = \frac{1 + \cos(2x)}{2} \). So, \( \sin^4 3t \cos^4 3t = (\sin^2 3t \cos^2 3t)^2 \).

Simplify Using Compound Identity

Since \( \sin^2 3t = \frac{1 - \cos(6t)}{2} \) and \( \cos^2 3t = \frac{1 + \cos(6t)}{2} \), we combine them: \( \sin^2 3t \cos^2 3t = \frac{1}{4}(1 - \cos^2(6t)) = \frac{1}{4}(\frac{1 - \cos(12t)}{2}) = \frac{1}{8}(1 - \cos(12t)) \).

Square the Expression

Now, square the result from Step 3: \(\left(\frac{1}{8}(1 - \cos(12t))\right)^2 = \frac{1}{64} (1 - 2\cos(12t) + \cos^2(12t)) \).

Integrate Term by Term

Integrate each part separately: \[ \int \frac{1}{64} (1 - 2\cos(12t) + \cos^2(12t)) dt = \frac{1}{64} \left( t - \frac{2}{12} \sin(12t) + \int \frac{1 + \cos(24t)}{2} dt \right) \].

Complete the Integration of Cosine Squared

The term \( \int \frac{1 + \cos(24t)}{2} dt \) can be broken into \( \frac{1}{2} t + \frac{1}{48} \sin(24t) \). So, the final integral becomes: \( \frac{1}{64} \left( t - \frac{1}{6} \sin(12t) + \frac{t}{2} + \frac{1}{48} \sin(24t) \right) + C \).

Simplify the Expression

Combine the like terms to simplify the expression: \( \frac{t}{128} + \frac{t}{128} - \frac{1}{384} \sin(12t) + \frac{1}{3072} \sin(24t) + C \). Therefore, the integral is: \( \frac{t}{32} - \frac{1}{192} \sin(12t) + \frac{1}{3072} \sin(24t) + C \).

Key concepts

Trigonometric Identities

Trigonometric identities are powerful tools used to simplify complex trigonometric expressions. In integration, they help to transform complicated integrands into manageable forms.

Some of the fundamental identities include:

• Pythagorean Identity: \(\sin^2 x + \cos^2 x = 1\)
• Angle Sum and Difference Identities, e.g., \(\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b\)
• Double Angle Identities, discussed in more detail later on.

These identities are frequently employed to reduce higher powers of sine and cosine functions, as they can transform the integrand into a form that makes using standard integration techniques possible.

For instance, terms like \(\sin^4 3t\) and \(\cos^4 3t\) can be rewritten using the square of sine or cosine identities, thereby simplifying the integration process. These rewritten forms enable easier calculation as they lay the groundwork for more direct integration approaches.

Integration Techniques

Integration techniques are methods used to find antiderivatives or integrals of functions. For trigonometric functions, specific methods like trigonometric identities and substitutions are often necessary.

Some common integration techniques include:

• Substitution: Changing variables to simplify the integral, particularly useful when dealing with composite functions.
• Integration by Parts: Useful for products of functions, derived from the product rule for differentiation.
• Trigonometric Identities: Simplifying the integrand using well-known trigonometric identities.

In the original exercise, the presence of \(\sin^4 3t \cos^4 3t\) indicates the use of these identities is crucial. By transforming the original expression into a simpler form, the integral becomes less challenging to handle, paving the way for straightforward integration.

Double Angle Identity

The double angle identity is a key trigonometric identity that expresses functions of double angles in terms of single angles. For sine and cosine, these identities are:

• \(\sin(2x) = 2\sin(x)\cos(x)\)
• \(\cos(2x) = \cos^2(x) - \sin^2(x) = 2\cos^2(x) - 1 = 1 - 2\sin^2(x)\)

In integration problems like the one presented, these identities help in expressing higher powers like \(\sin^4 3t\) and \(\cos^4 3t\) into more tractable forms, facilitating integration.

By substituting these identities into the integrand, the problem is broken down into parts that can be individually integrated. This step is crucial because it simplifies terms like higher power sines and cosines using products or functions of angle \(t\) alone.

Trigonometric Substitution

Trigonometric substitution is a technique where trigonometric identities substitute algebraic expressions to simplify an integral.

This method often appears in integrals involving root expressions of the form \(\sqrt{a^2 - x^2}\), \(\sqrt{a^2 + x^2}\), or \(\sqrt{x^2 - a^2}\). However, its principles can also apply to transform periodic trigonometric expressions into more straightforward components.In the context of the original exercise, even though a literal trigonometric substitution (e.g., replacing \(x\) with \(a \sin(\theta)\)) is not practiced, understanding how these transformations reform the structure of integrals is beneficial.

In our step-by-step solution, trig identities acted as a form of substitution, replacing complex expressions with simpler trigonometric equivalents, making the overall integration manageable and efficient.