Problem 69 - Calculus Early Transcendentals

Question

Use the reduction formulas in a table of integrals to evaluate the following integrals. $$\int \tan ^{4} 3 y d y$$

Step-by-Step Solution

Answer

Question: Determine the integral of the function \(f(y) = \tan^4 3y\). Answer: The integral of the function \(f(y) = \tan^4 3y\) is given by \(\int \tan^4 3y dy = \frac{1}{3}\tan^2 3y \sec^2 3y - \frac{2}{9}\tan 3y + D\), where D is the constant of integration.

Step 1: Identify the power and use the reduction formula for the first time

We have an integral of the form \(\int \tan^4 3y dy\), where n = 4. Applying the reduction formula, we get: $$\int \tan^4 3y dy = \frac{1}{4 - 1}\tan^{4 - 2} 3y \sec^2 3y - \frac{4 - 2}{4 - 1} \int \tan^{4 - 2} 3y dy$$ $$\int \tan^4 3y dy = \frac{1}{3}\tan^2 3y \sec^2 3y - \frac{2}{3} \int \tan^2 3y dy$$

Step 2: Reduce the power further using the reduction formula and simplify

We still have an integral involving a power of the tangent function, so we'll apply the reduction formula again to the remaining integral: $$\int \tan^2 3y dy = \frac{1}{2 - 1}\tan^{2 - 2} 3y \sec^2 3y - \frac{2 - 2}{2 - 1} \int \tan^{2 - 2} 3y dy$$ $$\int \tan^2 3y dy = \tan^0 3y \sec^2 3y = \sec^2 3y$$ Now substitute this result back into our previous expression: $$\int \tan^4 3y dy = \frac{1}{3}\tan^2 3y \sec^2 3y - \frac{2}{3} \int \sec^2 3y dy$$

Step 3: Integrate the remaining term

We have the integral \(\int \sec^2 3y dy\) remaining, which can be directly integrated: $$\int \sec^2 3y dy = \frac{1}{3} \tan 3y + C$$ Substitute this result back into our expression: $$\int \tan^4 3y dy = \frac{1}{3}\tan^2 3y \sec^2 3y - \frac{2}{3}\left(\frac{1}{3}\tan 3y + C\right)$$

Step 4: Simplify and write the final answer

Now we'll simplify the expression: $$\int \tan^4 3y dy = \frac{1}{3}\tan^2 3y \sec^2 3y - \frac{2}{9}\tan 3y - \frac{2}{3}C$$ Let's denote the constant with a different letter, say \(D = \frac{2}{3}C\). So the final answer will be: $$\int \tan^4 3y dy = \frac{1}{3}\tan^2 3y \sec^2 3y - \frac{2}{9}\tan 3y + D$$

Key Concepts

Integral Calculus

Integral calculus is a major part of calculus concerned with finding the integral of a function. It helps in understanding quantities like area, volume, and accumulation.

There are two main types of integrals:

• Definite Integrals: Provide a number representing the signed area under a curve and between specified limits.
• Indefinite Integrals: Represent a family of functions and include a constant of integration, often noted as +C.

The concept of integration can be visualized as the reverse process to differentiation. This means where derivatives give the rate of change, integrals give the accumulated quantity.

When solving integrals involving trigonometric functions, such as \(\int \tan^4 3y dy\), we often use techniques like substitution, integration by parts, or special formulas such as reduction formulas. These techniques transform complicated integrals into simpler forms that are easier to evaluate.

Power of Tangent Function

The tangent function, denoted as \( \tan(x) \), is periodic and related to the sine and cosine functions. It is used frequently in trigonometry and calculus.

In integrals like \( \int \tan^4 3y \, dy \), we deal with high powers of the tangent function.

To work with the tangent function in calculus:

• Reduction Formula: For an integral of the form \( \int \tan^n(x)dx \), reduction formulas help break down higher powers into more manageable terms.
• Power Reduction Identities: Sometimes, trigonometric identities are used to reduce the power of functions, for example, expressing tangent in terms of sine and cosine.

The tangent's derivative is \( \sec^2(x) \), which directly relates to its occurrence in integrals. Understanding this can help in solving related integrals by recognizing patterns and relationships.

Integration Techniques

Solving complex integrals often requires sophisticated techniques to simplify calculations. These techniques aim to transform a challenging integral into one that is straightforward to solve.

Here are some common techniques:

• Reduction Formulas: These are particularly useful for functions raised to a power, like \( \tan^n(x) \). They help diminish the power progressively.
• Substitution: This involves changing variables to simplify the integration. For instance, substituting \( u = 3y \) can sometimes help manage multipliers and exponents.
• Direct Integration: Often pursued after reducing the more complex parts to simpler integrals, as in the case of \( \int \sec^2(x)dx = \tan(x) + C \).

When using these techniques, it's crucial to understand the problem's specific structure and apply the best method that simplifies the process effectively.