Question

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

Short answer

Question: Evaluate the integral \(\int \tan^{4}(3y) dy\). Answer: The integral \(\int \tan^{4}(3y) dy\) can be evaluated as \(\frac{1}{3} \tan^2(3y) \sec^2(3y) - \frac{2}{9} \tan(3y) + \frac{2}{3} y+ C\), where \(C\) is the constant of integration.

Step-by-step solution

Identify the reduction formula

First, we need to identify the appropriate reduction formula for the given integral. In our case, we have the integral \(\int \tan^{4}(3y) dy\). The reduction formula for \(\int \tan^n(x) dx\) when `n` is an even number is: $$\int \tan^n(x) dx = \frac{1}{n-1} \tan^{n-2}(x) \sec^2(x) - \frac{n-2}{n-1}\int \tan^{n-2}(x) dx$$

Apply the reduction formula

We apply the reduction formula to our given integral, with \(n=4\) and \(x = 3y\): $$\int \tan^{4}(3y) dy = \frac{1}{4-1} \tan^{4-2}(3y) \sec^2(3y)- \frac{4-2}{4-1} \int \tan^{4-2}(3y) dy$$ Simplify the expression: $$\int \tan^{4}(3y) dy = \frac{1}{3} \tan^2(3y) \sec^2(3y) - \frac{2}{3} \int \tan^2(3y)dy$$

Use a trigonometric identity for \(\tan^2(x)\)

We will now need to use the trigonometric identity \(\tan^2(x) = \sec^2(x) - 1\). Substitute this identity into the integral: $$\int \tan^{4}(3y) dy = \frac{1}{3} \tan^2(3y) \sec^2(3y) - \frac{2}{3}\int(\sec^2(3y) - 1) dy$$

Separate the integral and integrate

Now separate the integral and integrate each part individually: $$\int \tan^{4}(3y) dy = \frac{1}{3} \tan^2(3y) \sec^2(3y) - \frac{2}{3} \int \sec^2(3y) dy + \frac{2}{3}\int dy$$ Integrating each part, we get: $$\int \tan^{4}(3y) dy = \frac{1}{3} \tan^2(3y) \sec^2(3y) - \frac{2}{3} (\frac{1}{3}\tan(3y) + C_1) + \frac{2}{3}(y+C_2)$$

Simplify the final result

Combine constant terms and simplify the final result: $$\int \tan^{4}(3y) dy = \frac{1}{3} \tan^2(3y) \sec^2(3y) - \frac{2}{9} \tan(3y) + \frac{2}{3} y+ C$$ where \(C = C_1 + C_2\) is the constant of integration.

Key concepts

Integral Calculus

Integral calculus is a fundamental concept in mathematics that is concerned with finding the total accumulation of quantities. Unlike differential calculus, which focuses on rates of change, integral calculus revolves around summation. The basic idea is to measure the area under a curve, represented by a function, over a specific interval.

In the context of integrals, there are two major types: definite and indefinite. While a definite integral has upper and lower limits which help calculate a precise area, an indefinite integral focuses on finding a general form of antiderivatives, often resulting in a function plus a constant of integration, denoted as \(C\).

• Definite Integral: Provides a numerical value representing the area between the curve of a function and the x-axis on a graph over a specific interval.
• Indefinite Integral: Returns a function that describes the antiderivative, which can account for any value through addition of the integration constant \(C\). This forms a family of curves, shifted vertically depending on different values of \(C\).

Mastering integral calculus is crucial for analyzing physical phenomena involving accumulation, such as finding areas, volumes, and solving differential equations.

Trigonometric Integrals

Trigonometric integrals involve the integration of functions containing trigonometric expressions. These are essential when dealing with periodic functions in various fields like physics, engineering, and signal processing. Common trigonometric functions include sine, cosine, tangent, and their reciprocals.

For example, integrating powers of trigonometric functions, such as \(\int \tan^n(x) dx\), often requires special strategies, including the use of identities and reduction formulas. These allow simplification and step-by-step integration to handle the complexities of the functions involved.

• Whenever encountering higher powers of trigonometric functions, it's useful to express some using Pythagorean identities, e.g., \(\tan^2(x) = \sec^2(x) - 1\), which helps to simplify and reduce integrals to a more manageable form.
• Other identities like \(\sin^2(x) + \cos^2(x) = 1\) can also aid in transforming the integrals.

These integrals can sometimes be challenging but practicing the application of identities and reduction formulas helps build comprehension and reduce errors during calculations.

Reduction Formula Application

The application of reduction formulas is a strategic technique used to simplify complex integrals, specifically those involving powers of functions like trigonometric functions. Reduction formulas help turn a difficult problem into easier and more manageable steps.

In our exercise, the integral \(\int \tan^4(3y) dy\) is simplified using a reduction formula designed for \(\int \tan^n(x) dx\) where \(n\) is an even number. This particular formula breaks down higher powers by converting them into lower powers accommodated with subsequent integrations.

• The formula: \[\int \tan^n(x) dx = \frac{1}{n-1} \tan^{n-2}(x) \sec^2(x) - \frac{n-2}{n-1}\int \tan^{n-2}(x) dx\]
• Each step reduces the power of the tangent function, leading to simplified integrals easier to solve.
• Utilizing trigonometric identities further simplifies procedures by breaking squares of functions into more straightforward components, as seen when substituting \(\tan^2(x) = \sec^2(x) - 1\).

Practicing reduction formula techniques can significantly aid students in handling otherwise convoluted integrals, preparing them for more advanced problem-solving scenarios.