Problem 82 - Calculus: Graphical, Numerical, Algebraic

Question

Trigonometric Substitution Suppose \(u=\tan ^{-1} x\) (a) Use the substitution \(x=\tan u, d x=\sec ^{2} u d u\) to show that \(\int \frac{d x}{1+x^{2}}=\int 1 d u\) (b) Evaluate \(\int 1 d u\) to show that \(\int \frac{d x}{1+x^{2}}=\tan ^{-1} x+C\)

Step-by-Step Solution

Answer

\(\int \frac{dx}{1 + x^2} = \tan^{-1}x + C\)

Step 1: Substitute \(x\) with \(\tan u\)

By replacing \(x\) with \(\tan u\) and \(dx\) with \(\sec^{2}u du\), the expression changes to: \(\int \frac{\sec^{2} u du}{1 + \tan^{2} u}\).

Step 2: Simplify using trigonometric identity

Trigonometric identities tell us that \(1 + \tan^{2} u = sec^{2} u\). Incorporating this into the integral simplifys the expression to: \(\int \frac{\sec^{2} u du}{\sec^{2} u}\)

Step 3: Simplify the fraction

The denominator and the numerator of the integrand are the same, so the fraction can be simplified to: \(\int du\).

Step 4: Calculate the integral

After integration, the expression changes to: \(u + C\).

Step 5: Back substitution

Last step is to replace \(u\) with its original expression \(\tan^{-1}x\), which yields the final result: \(\int \frac{dx}{1 + x^2} = \tan^{-1}x + C\).

Key Concepts

Indefinite Integrals

Indefinite integrals, often encountered in calculus, are a way of reversing the process of differentiation. Essentially, they represent the collection of all the antiderivatives of a given function. When we speak of integrating a function indefinitely, we are concerned with finding a function whose derivative gives us the original function.

The notation \(\int f(x)dx\) represents the indefinite integral of \(f(x)\). One of the crucial aspects to remember is the 'constant of integration', denoted by \(C\), which emphasizes that there are infinitely many antiderivatives, each differing by a constant. For example, the indefinite integral of \(\frac{dx}{1+x^2}\) results in \(\tan^{-1}x + C\), where \(C\) signifies that any constant added to \(\tan^{-1}x\) still yields a valid antiderivative.

Trigonometric Identities

Trigonometric identities are fundamental relationships between trigonometric functions that are true for all values within their domains. These identities are significant in solving complex trigonometric equations and expressions, and serve as powerful tools in integration.

Common trigonometric identities include Pythagorean identities, such as \(1 + \tan^2 u = \sec^2 u\), which plays a critical role in this exercise. Utilizing such identities allows us to simplify expressions into a form that can be easily integrated. In the given problem, by substituting \(\tan u\) for \(x\), we have to simplify the integral using our knowledge of these identities to progress towards the solution.

Integration Techniques

Integration techniques encompass a variety of methods used to carry out non-straightforward integrals. One such technique is trigonometric substitution, which employs trigonometric identities to simplify integrals involving square roots and quadratic expressions.

In the exercise, we see the application of this method, where \(x\) is substituted with \(\tan u\) and the resulting integrand is simplified using a trigonometric identity. By recognizing which substitution to use and how it transforms the integral, students can navigate through more complex scenarios effectively. Other integration techniques include integration by parts, partial fraction decomposition, and using inverse trigonometric functions as substitutions, to name a few.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as \(\tan^{-1}x\) or arctangent, are used to find angles when given the values of the trigonometric functions. These functions also emerge as results of certain integrals and are essential to understand while working with trigonometric substitution.

When incorporating inverse trigonometric functions within integrals, as seen in the exercise, it's important to remember their derivatives. For example, the derivative of \(\tan^{-1}x\) is \(\frac{1}{1+x^2}\), which directly ties into our integral problem. The integral of \(\frac{1}{1+x^2}dx\) yields \(\tan^{-1}x\), demonstrating the close relationship between differentiation and integration while solving calculus problems.