Problem 6 - Calculus: Early Transcendental Functions

Question

Evaluate the indicated integral. $$\int \sec ^{2} x \sqrt{\tan x} d x$$

Step-by-Step Solution

Answer

\(\frac{2}{3} \tan ^{3/2} x + C\)

Step 1: Substitution

Let \(u = \tan x\). Then the differential \(du\) is \(du = \sec^2 x \, dx\). This substitution transforms the original integral into \(\int u^{1/2} \, du\).

Step 2: Solve the New Integral

To solve \(\int u^{1/2} \, du\) is straightforward application of the power rule. The power rule states that the integral of \(x^n\) is \(\frac{1}{n+1}x^{n+1}\), so the integral becomes \(\frac{2}{3} u^{3/2}\).

Step 3: Replace \(u\) with \(\tan x\)

Substitute \(u = \tan x\) back into the integral: \(\frac{2}{3} u^{3/2} = \frac{2}{3} \tan ^{3/2} x\).

Step 4: Determine the Constant of Integration

Don't forget to determine the constant of integration after carrying out the integral. Thus, the final result is \(\frac{2}{3} \tan ^{3/2} x + C\), where \(C\) is the constant of integration.

Key Concepts

Integral Substitution

Integral substitution is a method used to simplify complex integrals. It often involves substituting part of the original integrand (the function being integrated) with a new variable. This transformation makes the integral easier to evaluate. In the given exercise, we use substitution by setting \(u = \tan x\). This choice helps because the derivative of \(\tan x\) is \(\sec^2 x\), which already appears in our integral.

• Finding \(du\): Once we define \(u\), we find the differential \(du = \sec^2 x \, dx\). This replacement directly matches part of our original integrand.
• Transforming the integral: The substitution changes the original integral \(\int \sec^2 x \sqrt{\tan x} \, dx\) to \(\int u^{1/2} \, du\).

This approach simplifies the problem into a more manageable form and helps isolate the variable in a simple integral form.

Power Rule for Integration

The power rule is a fundamental principle in calculus for integrating functions of the form \(x^n\). According to this rule, the integral of \(x^n\) with respect to \(x\) can be found using \[ \int x^n \, dx = \frac{1}{n+1} x^{n+1} + C, \] where \(n eq -1\). This rule is easy to apply and particularly useful for polynomials and polynomial-like functions.
In our exercise, after substitution, the integral becomes \(\int u^{1/2} \, du\). We apply the power rule by setting \(n = \frac{1}{2}\). After integration, this results in \(\frac{2}{3} u^{3/2}\).

• Simplifying the expression: The rule transforms complex expressions to simpler forms, allowing us to solve integrals by handling powers of variables straightforwardly.
• Including the constant of integration: Remember that when calculating indefinite integrals, we add a constant \(C\). It accounts for any constant differences between antiderivatives.

Definite and Indefinite Integrals

Integrals can be classified as either definite or indefinite.

• Indefinite Integrals: These integrals lack specific boundaries and lead to a family of functions characterized by a constant of integration \(C\). They are represented as \(\int f(x) \, dx = F(x) + C\). In our solution, since no limits of integration are given, the result *\(\frac{2}{3} \tan^{3/2} x + C\)* represents an indefinite integral.
• Definite Integrals: These are calculated over a specific interval \([a, b]\) and yield a numerical result \(\int_{a}^{b} f(x) \, dx\), describing the area under the curve \(f(x)\) from \(x = a\) to \(x = b\). Although not directly required here, it's important to understand how they contrast from indefinite integrals.

Understanding both types is crucial because they serve different purposes in calculus and applications, such as solving differential equations or finding areas and volumes.