Problem 22 - Calculus of a Single Variable

Question

Evaluating a Definite Integral In Exercises \(21-32\) evaluate the definite integral. $$ \int_{0}^{\sqrt{2}} \frac{1}{\sqrt{4-x^{2}}} d x $$

Step-by-Step Solution

Answer

\(\pi/4\)

Step 1: Identify the integral form

The integral \(\int \frac{1}{\sqrt{4-x^2}} dx\) resembles the form of \(\int \frac{1}{\sqrt{a^2-x^2}} dx\) which standard result is \(\sin^{-1}(x/a) + C\), where \(a\) is a constant.

Step 2: Apply the integral formula

Applying the formula, we get \(\sin^{-1}(x/2) + C\).

Step 3: Evaluate the definite integral

The definite integral from \(0\) to \(\sqrt{2}\) is evaluated by subtracting the antiderivative at the lower limit from the antiderivative at the upper limit, which is \(\sin^{-1}(\sqrt{2}/2) - \sin^{-1}(0)\).

Key Concepts

Trigonometric Substitution

When you encounter an integral involving expressions like \( \sqrt{a^2 - x^2} \), trigonometric substitution can be very useful. This method helps simplify the integration by transforming the variables to match trigonometric identities.
Here’s why it’s effective:

• The expression \( a^2 - x^2 \) hints at the Pythagorean identity, \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
• Setting \( x = a \sin(\theta) \) transforms \( \sqrt{a^2 - x^2} \) into \( a\cos(\theta) \), a simpler form to integrate.

This substitution allows us to use our understanding of trigonometric functions and their inverses to handle more complex integrals intuitively.
In our original exercise, recognizing this similarity led to using the standard inverse sine formula.

Antiderivatives

The antiderivative, or the indefinite integral, is essentially the "reverse" of differentiation. It represents a family of functions whose derivative gives the original function.

For instance, in our problem, the antiderivative of \( \frac{1}{\sqrt{4-x^2}} \) was recognized as \( \sin^{-1}(x/2) + C \), where \( C \) is the constant of integration. This stems from the standard form of \( \int \frac{1}{\sqrt{a^2-x^2}} dx \) which equals \( \sin^{-1}(x/a) + C \). Recognizing and applying these known antiderivatives streamlines the integration process significantly.
Understanding antiderivatives is a key step in solving integrals, especially definite integrals that require evaluating limits after finding the antiderivative.

Integral Formulas

Integral formulas serve as shortcuts to finding antiderivatives, especially when dealing with complex functions. These formulas are derived from well-known integrals of fundamental functions.
Some important integral formulas include:

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) for \( n eq -1 \)
• \( \int e^x \, dx = e^x + C \)
• \( \int \sin(x) \, dx = -\cos(x) + C \)

In the solution to our problem, the integral \( \int \frac{1}{\sqrt{4-x^2}} dx \) was aligned with the known form \( \int \frac{1}{\sqrt{a^2-x^2}} dx = \sin^{-1}(x/a) + C \). This allowed us to directly write the antiderivative, speeding up the calculation.
Knowing and keeping these formulas handy is invaluable for efficiently evaluating both definite and indefinite integrals.