Problem 25 - Essential Calculus

Question

Find the integral involving secant and tangent. $$ \int \sec ^{2} x \tan x d x $$

Step-by-Step Solution

Answer

The result of the integral is \(\frac{1}{2}\tan^2x + C\).

Step 1: Identifying a substitution

Note that the derivative of \(\tan x\) is \(\sec^2x\). Hence, let's substitute \(u = \tan x\). Now differentiate \(u\) with respect to \(x\) to get \(du = \sec^2x dx\).

Step 2: Substituting into the integral

Substitute \(u = \tan x\) and \(du = \sec^2x dx\) into the integral. This simplifies the integral to \(\int u du\).

Step 3: Solving the Integral

The integral of \(u\) with respect to \(u\) is \(\frac{1}{2}u^2 + C\), where C is the constant of integration.

Step 4: Substituting back

Now substitute \(\tan x\) back in for \(u\) to get the final answer. Thus, the result of the integral is \(\frac{1}{2}\tan^2x + C\).

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify certain types of integrals, especially those involving square roots or trigonometric functions like secant, tangent, sine, and cosine.
It relies on the identities and properties of trigonometric functions to transform the integral into a simpler one. This is often done by substituting a trigonometric function with a variable to simplify the expression.
In the problem given, we use substitution by letting \( u = \tan x \), as the derivative of \( \tan x \), which is \( \sec^2 x \), appears in the integral.

• First, you explicitly state your substitution: \( u = \tan x \).
• Then, differentiate \( u \) with respect to \( x \), giving \( du = \sec^2 x \, dx \).

Using this substitution makes the integral more straightforward and manageable outside of the trigonometric context, transforming it into
\( \int u \, du \). This kind of substitution simplifies the computation significantly, particularly for integrals involving secants and tangents.

Definite and Indefinite Integrals

In calculus, integrals come in two main varieties: definite and indefinite.
Understanding the difference between them is crucial for solving integrals correctly.
An indefinite integral, as shown in the exercise, represents a family of functions and includes a constant of integration \( C \).

• Indefinite integrals are written without limits and result in a general antiderivative: \( \int f(x) \, dx = F(x) + C \).
• The integration process for indefinite integrals finds any function \( F(x) \) whose derivative is \( f(x) \).

In the provided solution, \( \int \sec^2 x \tan x \, dx \) falls under indefinite integrals because no upper or lower bounds are set.
Its solution is \( \frac{1}{2} \tan^2 x + C \).On the other hand, definite integrals have specific limits and evaluate the net area under the curve.
They are usually expressed as \( \int_{a}^{b} f(x) \, dx \) and result in a numerical value, not a function.

Differential Calculus

Differential calculus deals primarily with the concept of the derivative, which represents the rate of change of a function.
It is an essential part of calculus that provides the basis for trigonometric substitution and helps in determining derivatives needed for solving integrals.

• The derivative of a function, such as \( \tan x \), is derived using rules from differential calculus. Here, \( \frac{d}{dx}(\tan x) = \sec^2 x \).
• This derivative transforms the original integral \( \int \sec^2 x \tan x \, dx \) into a much simpler form with substitution.

Differential calculus allows us to identify parts of the integral that match up perfectly with derivatives of well-known functions.
This knowledge is crucial for employing substitutions like \( u = \tan x \) in transforming and simplifying integrals.
Understanding these derivatives enables us to maneuver through complex integrals with ease, cementing the dependence of integral calculus on its differential counterpart.