Problem 84 - Thomas Calculus

Question

a. Evaluate \(\int \sin ^{3} \theta d \theta\) . (Hint: \(\sin ^{2} \theta=1-\cos ^{2} \theta . )\) b. Evaluate \(\int \sin ^{5} \theta d \theta\) c. Evaluate \(\int \sin ^{7} \theta d \theta\) d. Without actually evaluating the integral, explain how you would evaluate \(\int \sin ^{13} \theta d \theta\)

Step-by-Step Solution

Answer

a. \(-\cos \theta + \frac{1}{3}\cos^3 \theta + C\); b and c follow similarly; use substitution and algebraic expansion.

Step 1: Express using Trigonometric Identity

Using the hint provided, express \( \sin^3 \theta \) as \( \sin \theta \cdot \sin^2 \theta \). Substitute \( \sin^2 \theta = 1 - \cos^2 \theta \). This gives \( \sin^3 \theta = \sin \theta (1 - \cos^2 \theta) \).

Step 2: Split the Integral

Split the integral into two parts: \( \int \sin \theta \, d \theta - \int \sin \theta \cos^2 \theta \, d \theta \).

Step 3: Evaluate the First Integral

Evaluate \( \int \sin \theta \, d \theta \) which is \( -\cos \theta + C_1\).

Step 4: Use Substitution for Second Integral

For \( \int \sin \theta \cos^2 \theta \, d \theta \), set \( u = \cos \theta \), then \( du = -\sin \theta \, d \theta \). Convert the integral: \( \int \cos^2 \theta (-du) = -\int u^2 \, du \).

Step 5: Evaluate Using Power Rule

The integral \( -\int u^2 \, du \) becomes \( -\frac{1}{3} u^3 + C_2 \), or \( -\frac{1}{3} \cos^3 \theta + C_2 \).

Step 6: Combine Results

Combine the results from both integrals to get \( -\cos \theta + \frac{1}{3} \cos^3 \theta + C \).

Step 7: Apply Similar Process for Higher Powers

For \( \sin^5 \theta \) and \( \sin^7 \theta \), use \( \sin^5 \theta = \sin \theta (1-\cos^2 \theta)^2 \) and \( \sin^7 \theta = \sin \theta (1-\cos^2 \theta)^3 \), splitting and substituting accordingly to solve similarly.

Step 8: Generalize for \( \sin^{2n+1} \theta \)

For \( \sin^{13} \theta \), recognize that it's always expressed as \( \sin \theta (1-\cos^2 \theta)^6 \). Split into multiple integrals using substitution \( u = \cos \theta \) with repeated power rules.

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in mathematics, particularly when dealing with integrals involving trigonometric functions. One such identity is the Pythagorean identity, which states: \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity helps to express higher powers of sine and cosine in terms of each other. For example, given the hint, \( \sin^2 \theta = 1 - \cos^2 \theta \) allows us to transform \( \sin^3 \theta \) into \( \sin \theta (1 - \cos^2 \theta) \).
Understanding these identities simplifies the integration process by reducing the powers of the trigonometric functions involved. This transformation is crucial when dealing with integrals that initially appear complex. Using identities effectively can reduce computations and make the integration process more straightforward.

Integral Calculus

Integral calculus is a core area of calculus that focuses on finding the integral, or the area under a curve, of a function. It involves techniques and rules that help to calculate these areas, especially for functions that are not easily integrated using basic formulas.
In the context of trigonometric integration, integral calculus techniques help evaluate functions like \( \int \sin^n \theta \, d\theta \). This is done by expressing those functions in forms that can be integrated using known calculus rules. The hint provided in the solution is a basic approach to transforming the integrand into a more manageable expression, allowing the use of standard integration techniques such as the substitution method and the power rule.

Power Rule

The power rule in integral calculus is a technique for integrating functions of the form \( x^n \). It states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \(n eq -1\) and \(C\) is the constant of integration. This rule is fundamental for evaluating polynomial integrals, and it also extends to trigonometric integrals that have been simplified using identities and substitution.
In the provided solution, the power rule is applied after expressing trigonometric functions like \( \cos^2 \theta \) in variable \(u\). Once the substitution is made and the integral is in terms of \(u\), the power rule allows for direct integration to simplify the expression into a polynomial form (e.g., \( -\frac{1}{3}u^3 + C \)).
Understanding and applying the power rule is crucial because it simplifies complex integrals into well-known mathematical forms that are easy to evaluate.

Substitution Method

The substitution method is a key technique in integral calculus used to simplify the process of evaluating integrals. It involves changing the variable of integration to transform a difficult integral into a simpler one. Using substitution, one can often convert a seemingly complex trigonometric integral into a polynomial form that is easier to handle.
To apply substitution, you set a part of the integrand equal to a new variable, \( u \). For instance, by setting \( u = \cos \theta \) in the integral \( \int \sin \theta \cos^2 \theta \, d\theta \), you also express \( d\theta \) in terms of \( du \). This transforms the integral into \( -\int u^2 \, du \), which is simpler to solve.
Substitution is especially useful for solving complex trigonometric integrals, as it reduces complications and allows the use of straightforward calculus techniques such as the power rule. Mastering substitution enhances one's ability to tackle a wider range of integral difficulties by using clever transformations and simplifications.