Problem 37 - Calculus Early Transcendentals

Question

Integrals involving tan \(x\) and sec \(x\) Evaluate the following integrals. $$\int \sec ^{2} x \tan ^{1 / 2} x d x$$

Step-by-Step Solution

Answer

Question: Evaluate the integral $$\int \sec ^{2} x \tan ^{1 / 2} x d x$$ Answer: $$\frac{2}{3} (\tan x)^{3/2} + C$$

Step 1: Choose a substitution

Let's do a substitution to simplify the integral. Let: $$u = \tan x$$ Now find the derivative of \(u\) with respect to \(x\): $$\frac{d u}{d x} = \frac{d (\tan x)}{dx} = \sec^2 x$$ Now, solve for \(dx\): $$d x = \frac{du}{\sec^2 x}$$

Step 2: Substitute u into the integral

Rewrite the integral in terms of \(u\) by making substitutions: $$\int \sec ^{2} x \tan ^{1 / 2} x d x = \int u^{1/2} \cdot du$$

Step 3: Integrate with respect to u

Now, integrate the simplified integral with respect to \(u\): $$\int u^{1/2} \cdot du = \frac{2}{3} u^{3/2} + C$$

Step 4: Substitute back for x

Finally, substitute back for \(x\) by replacing \(u\) with \(\tan x\): $$\frac{2}{3} u^{3/2} + C = \frac{2}{3} (\tan x)^{3/2} + C$$ The evaluated integral is: $$\int \sec ^{2} x \tan ^{1 / 2} x d x = \frac{2}{3} (\tan x)^{3/2} + C$$

Key Concepts

Substitution Method

The Substitution Method is a powerful tool in calculus to simplify integration, particularly when handling complex integrals like the given example. The ultimate goal is to transform the original variable, to make the integral easier to solve. This is done by substituting part of the integral with a new variable, typically labeled 'u'. By identifying a suitable substitution, such as setting \( u = \tan x \), we can effectively transform the original function. Recognizing the derivative of \( an x \), which is \( \sec^2 x \), allows us to express \( dx \) in terms of \( du \). This translates the integral from one involving trigonometric functions to a simpler algebraic expression.

• Choose a substitution - Opt for a part of the function that simplifies the integral.
• Derive the new variable - Differentiate the substitution equation to express \( dx \) or \( du \).
• Substitute and integrate - Replace and simplify the integral, making it more straightforward to integrate.

Learning the skill of choosing useful substitutions can simplify many complex integrals, highlighting the versatility and practicality of the substitution method.

Trigonometric Integrals

Trigonometric Integrals involve the integration of products of trigonometric functions. These types of integrals can often be tricky due to the functions' periodic and oscillating nature. However, understanding specific identities and strategies makes them much more manageable.For the exercise involving \( \sec^2 x \tan^{1/2} x \), knowledge of trigonometric identities and relationships is key. For instance:

• The derivative of \( \tan x \) is \( \sec^2 x \), which simplifies integrations involving these functions drastically.
• Identities such as \( \sin^2 x + \cos^2 x = 1 \) aid in transforming functions to more integrable forms when directly applicable.

Often, the challenge is recognizing the combination of trigonometric relationships that will simplify the integration process. By employing the right trigonometric transformations, integration can proceed smoothly even for seemingly complicated expressions.

Definite and Indefinite Integrals

Differentiating between definite and indefinite integrals is fundamental in calculus. An indefinite integral, like the one solved in the exercise, refers to the antiderivative of a function and is expressed with a constant of integration, \( C \). This constant accounts for any vertical shift in the function and indicates that there are infinitely many antiderivatives for a given integrand.Definite integrals, on the other hand, have specific limits of integration and calculate the net area under the curve of the function within these bounds. Unlike indefinite integrals, the result of a definite integral is a real number, representing this area.

• Indefinite integrals - Represent antiderivatives and include the integration constant \( C \).
• Definite integrals - Provide the exact area under the curve, evaluated between two specific limits.

Understanding these integrals and when to apply them is crucial for solving real-world problems, from physics to engineering, as they represent accumulation processes, such as distance or total cost.