Problem 2 - Calculus of a Single Variable

Question

Finding an Indefinite Integral In Exercises \(1-20\) , find the indefinite integral. $$ \int \frac{d x}{\sqrt{1-4 x^{2}}} $$

Step-by-Step Solution

Answer

The result of the integral \( \int \frac{dx}{\sqrt{1-4x^2}} \) is \( 2 \arcsin(2x) + C \).

Step 1: Identify the integral form

The integral is clearly of the form \( \int \frac{dx}{\sqrt{a^2 - x^2}} \) where \(a^2\)=1 and the coefficient of \(x^2\) is -4.

Step 2: Make the coefficient of \(x^2\) to be 1

Let \( x = \frac{u}{2} \) such that the coefficient of \(x^2\) is 1. So, the integral becomes \( \int \frac{2du}{\sqrt{1 - u^2}} \).

Step 3: Evaluate the Integral

Now, apply the standard formula \( \int \frac{du}{\sqrt{a^2 - u^2}} = \arcsin(\frac{u}{a}) + C \), for the integral. When you apply the formula, you get \(2 \arcsin(u) + C\).

Step 4: Back substitution

Substitute \(u = 2x\), to back-substitute the value of \(u\) to get \(2 \arcsin(2x) + C \).

Key Concepts

Indefinite Integral

The concept of an indefinite integral is fundamental in calculus. When we talk about indefinite integrals, we are referring to the antiderivative of a function. Unlike definite integrals, which have limits and result in a numerical value, indefinite integrals do not have specified bounds and provide a family of functions.
To find the indefinite integral, denoted by the integral symbol without upper and lower limits,

• We seek the original function whose derivative gives us the function under the integral sign.
• Constant of integration (usually represented as "C") is always added to account for the fact that there are infinitely many functions of the same derivative, differing by a constant.

In our case, \[\int \frac{dx}{\sqrt{1-4x^2}} \]represents an indefinite integral, as it lacks the limits of integration. We aim to find the antiderivative of the given expression without evaluating it over a specific interval.

Integration Techniques

To solve integrals like \[\int \frac{dx}{\sqrt{1-4x^2}},\]we often apply various integration techniques. These methods simplify the integration process, making it easier to find the antiderivative. Common integration techniques include:

• Substitution: Often helpful when dealing with composite functions. Here, we use substitution to simplify the integral by letting \(x = \frac{u}{2}\) to make the integral easier to solve.
• Integration by Parts: Useful when the integral is a product of functions, though not directly applicable in this problem.
• Partial Fractions: This technique splits a complex rational function into simpler fractions, but is not used here as our integrand is not a rational expression.

In our exercise, substitution is effective to transform the expression into a simpler form. By setting \(x = \frac{u}{2}\), we changed the coefficient of \(x^2\) in the radical to 1. This simplifies the given integral to a standard form more straightforward to evaluate.

Trigonometric Substitution

Trigonometric substitution is a special integration technique useful when dealing with integrals involving square roots with quadratic expressions. It often simplifies the process by transforming algebraic expressions into trigonometric ones, which are easier to integrate.
For the integral \[\int \frac{2du}{\sqrt{1-u^2}},\]we used the fact that the integral resembles the form \(\int \frac{du}{\sqrt{a^2 - u^2}}.\)This indicates a natural trigonometric substitution:

• The standard integral for this form is \(\arcsin\left(\frac{u}{a}\right) + C.\)
• In our case, we directly used this standard formula after substitution, identifying \(a\) as 1.

Upon substitution with \(x = \frac{u}{2}\), we ultimately find that our solution involves the arcsine function, giving us \(2 \arcsin(2x) + C.\)This showcases how trigonometric substitution and knowledge of trigonometric identities simplifies seemingly complex integrals.