Problem 31 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(23-32\). $$ \int_{\pi / 6}^{\pi / 3} \cot ^{3} x d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( 2 - \ln(\sqrt{3}) \).

Step 1: Rewrite the integrand

The integral given is \( \int_{\pi / 6}^{\pi / 3} \cot^3 x \, dx \). Instead of integrating \( \cot^3 x \) directly, rewrite it using trigonometric identities: \( \cot x = \frac{\cos x}{\sin x} \). With this identity, we can express \( \cot^3 x \) as \( \frac{\cos^3 x}{\sin^3 x} \).

Step 2: Simplify using trigonometric identity

Recognize that \( \cot^3 x = \left( \frac{\cos x}{\sin x} \right)^3 = \frac{\cos^2 x}{\sin^3 x} \cdot \cos x \). Use the identity \( 1 - \sin^2 x = \cos^2 x \) to substitute: \( \frac{(1 - \sin^2 x)}{\sin^3 x} \cdot \cos x \).

Step 3: Substitution

To make integration easier, use the substitution \( u = \sin x \), then \( du = \cos x \, dx \). Change the bounds of the integral: when \( x = \pi/6 \), \( u = \sin(\pi/6) = 1/2 \), and when \( x = \pi/3 \), \( u = \sin(\pi/3) = \sqrt{3}/2 \). The integral becomes \( \int_{1/2}^{\sqrt{3}/2} \frac{1 - u^2}{u^3} \, du \).

Step 4: Split the integral

Split the integral into two simpler integrals: \( \int_{1/2}^{\sqrt{3}/2} \frac{1}{u^3} \, du - \int_{1/2}^{\sqrt{3}/2} \frac{u^2}{u^3} \, du \), which simplifies to \( \int_{1/2}^{\sqrt{3}/2} u^{-3} \, du - \int_{1/2}^{\sqrt{3}/2} u^{-1} \, du \).

Step 5: Integrate each part

Integrate each part separately: \( \int_{1/2}^{\sqrt{3}/2} u^{-3} \, du = \left[ \frac{u^{-2}}{-2} \right]_{1/2}^{\sqrt{3}/2} \) and \( \int_{1/2}^{\sqrt{3}/2} u^{-1} \, du = [ \ln |u| ]_{1/2}^{\sqrt{3}/2} \).

Step 6: Evaluate the definite integrals

Evaluate \( \left[ \frac{u^{-2}}{-2} \right]_{1/2}^{\sqrt{3}/2} \) to get \( \left( -\frac{1}{2(3/4)} + \frac{1}{2(1/4)} \right) = -2 + 4 = 2 \). Evaluate \([ \ln |u| ]_{1/2}^{\sqrt{3}/2} \) to get \( \ln(\sqrt{3}/2) - \ln(1/2) = \ln(\sqrt{3}/2) + \ln(2) = \ln(\sqrt{3}) \).

Step 7: Combine results

Combine the results from the evaluated integrals to find the value of the integral: \( 2 - \ln(\sqrt{3}) \).

Key Concepts

Definite Integrals

A definite integral is essentially the net area under a curve within a specific interval. Unlike indefinite integrals, which do not have bounds, a definite integral has upper and lower limits of integration. This means:

• The integral \( \displaystyle \int_{a}^{b} f(x) \, dx \) is calculated over the interval \( [a, b] \).
• The result of a definite integral is a number, which represents the accumulated area between the function graph and the x-axis over the given limits.

Many integration techniques, including trigonometric substitution, are specifically useful for handling complicated integrands within definite integrals. When evaluating a definite integral, remember to:

• Ensure proper substitution of limits post-integration.
• Carefully consider the function's behavior across the interval.

Trigonometric Substitution

Trigonometric substitution is a method used to simplify integrals involving square roots and quadratic expressions. The idea is to substitute a trigonometric function (like sine, cosine, or tangent) to transform the integrand into a more easily integrable form. This method can often simplify complicated expressions, particularly those with:

• Squares and square roots.
• Trigonometric identities at play.

In our original solution, we used a substitution \( u = \sin x \), which helped to reduce the original complicated form of \( \cot^3 x \) into a rational function. This substitution not only simplified the expressions but also allowed us to switch the variable of integration effectively.

Trigonometric Identities

Trigonometric identities are fundamental tools in simplifying expressions and proving equations within trigonometric contexts. They serve as shortcuts and provide critical substitutions needed for integration. Some key identities include:

• \( \sin^2 x + \cos^2 x = 1 \)
• \( 1 - \sin^2 x = \cos^2 x \)
• \( \tan x = \frac{\sin x}{\cos x} \)

In the solution, we used the identity \( \cot x = \frac{\cos x}{\sin x} \) to reformulate \( \cot^3 x \) into an expression involving \( \sin x \) and \( \cos x \). This manipulation helps not only in writing the integrand in a more integrable form but also in using subsequent substitutions to further simplify the integration process.

Evaluation of Integrals

Evaluation of integrals, especially definite ones, often involves several steps, including rewriting, substituting, and finally integrating to find the solution. This process requires carefully:

• Choosing appropriate substitutions to simplify the integrand.
• Splitting complicated integrals into simpler parts when necessary.
• Integrating each part individually and combining the results.

In the given solution, the integral was rewritten using substitutions and identities to simplify it. We then split it into two more accessible integrals, accepted one by one. Post integration, the unique definite integral solution emerges from evaluating the integral's upper and lower bounds. This approach ensures accuracy in finding the net area or accumulated value defined by the limits.