Problem 28 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-28, perform the indicated integrations. \(\int \tan ^{3} x \sec ^{-1 / 2} x d x\)

Step-by-Step Solution

Answer

The integral evaluates to \(-\frac{2}{3} \sec^{3/2} x + 2 \sec^{1/2} x + C\).

Step 1: Rewrite in Known Forms

Begin by rewriting the given integral \( \int \tan ^{3} x \sec ^{-1 / 2} x \, dx \). Recall that \( \tan x = \frac{\sin x}{\cos x} \) and \( \sec x = \frac{1}{\cos x} \). Therefore, the integral becomes:\[ \int \frac{\sin^3 x}{\cos^3 x} \cdot \cos^{1/2} x \, dx = \int \frac{\sin^3 x}{\cos^{5/2} x} \, dx. \]

Step 2: Use Substitution

Let \( u = \cos x \), then \( du = -\sin x \, dx \) or \( \sin x \, dx = -du \).Substitute these into the integral to get:\[ -\int \frac{\sin^2 x \cdot \sin x}{u^{5/2}} \, du \]Using \( \sin^2 x = 1 - u^2 \), we have:\[ -\int \frac{(1-u^2) \sin x}{u^{5/2}} \, du. \]

Step 3: Simplify and Split the Integral

Continue simplifying:\[ \int (1-u^2) \cdot \sin x \, dx \text{ becomes } \int (1 - u^2) \cdot \frac{du}{u^{5/2}}. \]This splits into two separate integrals:\[ \int u^{-5/2} \, du - \int u^{-(1/2)} \, du. \]

Step 4: Perform the Integration

Independently integrate each part:\[ \int u^{-5/2} \, du = \frac{u^{-3/2}}{-3/2} = -\frac{2}{3} u^{-3/2}. \]\[ \int u^{-1/2} \, du = \frac{u^{1/2}}{1/2} = 2u^{1/2}. \]Thus, the combined integral is:\[ -\frac{2}{3} u^{-3/2} + 2u^{1/2} + C. \]

Step 5: Substitute Back

Replace the \( u \) with \( \cos x \):\[ -\frac{2}{3} (\cos x)^{-3/2} + 2(\cos x)^{1/2} + C. \]Simplify where possible:\[ -\frac{2}{3} \sec^{3/2} x + 2 \sec^{1/2} x + C. \]

Key Concepts

Trigonometric Integration

Trigonometric integration refers to the process of integrating functions that involve trigonometric functions such as sine (\( \sin x \)), cosine (\( \cos x \)), and tangent (\( \tan x \)). In this exercise, we're dealing with \( \tan^3 x \) and \( \sec^{-1/2} x \).

Trigonometric identities are essential tools here:

• \( \tan x = \frac{\sin x}{\cos x} \)
• \( \sec x = \frac{1}{\cos x} \)

By rewriting these, we convert complicated trigonometric expressions into more manageable forms. In our case, the integral \( \int \tan^3 x \sec^{-1/2} x \, dx \) becomes:\[ \int \frac{\sin^3 x}{\cos^{5/2} x} \, dx. \]This transformation allows us to apply other techniques, such as substitution, more seamlessly.

Substitution Method

The substitution method is a powerful technique for simplifying integrals by changing variables. Here, we substitute \( u = \cos x \), transforming the integration variable from \( x \) to \( u \).

When \( u = \cos x \), the derivative is \( du = -\sin x \, dx \), and therefore \( \sin x \, dx = -du \). Substitution allows us to rewrite the integral in terms of \( u \):\[ -\int \frac{\sin^2 x \cdot \sin x}{u^{5/2}} \, du \]

To complete the substitution, note \( \sin^2 x = 1 - u^2 \), transforming the integral further to:\[ -\int \frac{(1-u^2) \sin x}{u^{5/2}} \, du. \]Substitution simplifies complex expressions, making them more approachable for integration.

Integral Simplification

Simplifying an integral often involves breaking it down into separate, more manageable parts. Here, after substitution, we use algebra to split the integral:\[ \int (1-u^2) \cdot \frac{du}{u^{5/2}} = \int u^{-5/2} \, du - \int u^{-(1/2)} \, du. \]

By dividing the integral into two separate parts, each part can be approached individually. This makes the problem more straightforward, as each sub-integral corresponds to a basic power function form. Solving these basic forms requires applying simple integration rules:

• \( \int u^{-5/2} \, du \) corresponds to \( -\frac{2}{3} u^{-3/2} \)
• \( \int u^{-1/2} \, du \) corresponds to \( 2u^{1/2} \)

Performing integral simplification disentangles complex relationships within the integrand, resulting in a clearer path toward the solution.

Calculus Problem Solving

Calculus problem solving often involves creatively applying mathematical concepts to arrive at a solution. In this problem, using different techniques like rewriting, substitution, and simplification guides us through.

Once each part of the integral is handled separately:

• \( -\frac{2}{3} u^{-3/2} + 2u^{1/2} + C \)

We must reverse the substitution to return to the original variable:\[-\frac{2}{3} (\cos x)^{-3/2} + 2 (\cos x)^{1/2} + C. \]This produces the final solution:\[-\frac{2}{3} \sec^{3/2} x + 2 \sec^{1/2} x + C. \]

Problem solving in calculus is akin to piecing together a puzzle; different techniques and methods combine to offer a complete picture.