Problem 53 - Thomas Calculus

Question

Evaluate each integral in Exercises \(53-56\) by separating the fraction and using a substitution (if necessary) to reduce it to standard form. $$ \int \frac{1-x}{\sqrt{1-x^{2}}} d x $$

Step-by-Step Solution

Answer

\( \int \frac{1-x}{\sqrt{1-x^2}} \, dx = \sin^{-1}(x) - \sqrt{1-x^2} + C \).

Step 1: Separate the Fraction

The fraction \( \frac{1-x}{\sqrt{1-x^2}} \) can be separated into two parts: \( \frac{1}{\sqrt{1-x^2}} \) and \( \frac{-x}{\sqrt{1-x^2}} \). Thus, the integral becomes: \[ \int \frac{1-x}{\sqrt{1-x^2}} \, dx = \int \frac{1}{\sqrt{1-x^2}} \, dx - \int \frac{x}{\sqrt{1-x^2}} \, dx. \]

Step 2: Evaluate the First Integral

The first integral \( \int \frac{1}{\sqrt{1-x^2}} \, dx \) is a standard form. It is the derivative of \( \sin^{-1}(x) \), so: \[ \int \frac{1}{\sqrt{1-x^2}} \, dx = \sin^{-1}(x) + C_1, \] where \( C_1 \) is the constant of integration.

Step 3: Use Substitution for the Second Integral

For the second integral \( \int \frac{x}{\sqrt{1-x^2}} \, dx \), use the substitution \( u = 1 - x^2 \), which gives \( du = -2x \, dx \) or \( x \, dx = -\frac{1}{2} \, du \). Thus, the integral becomes: \[ -\int \frac{x}{\sqrt{1-x^2}} \, dx = \frac{1}{2} \int \frac{1}{\sqrt{u}} \, du. \]

Step 4: Evaluate the Substituted Integral

The substituted integral \( \frac{1}{2} \int \frac{1}{\sqrt{u}} \, du \) is the integral of \( u^{-\frac{1}{2}} \), which evaluates to: \[ \frac{1}{2}(2\sqrt{u}) + C_2 = \sqrt{u} + C_2. \] Substituting back \( u = 1-x^2 \), we get \( \sqrt{1-x^2} + C_2 \).

Step 5: Combine the Results

Combining the results from Steps 2 and 4, the original integral is: \[ \sin^{-1}(x) + C_1 - (\sqrt{1-x^2} + C_2). \] Simplifying, we drop the constants of integration: \[ \sin^{-1}(x) - \sqrt{1-x^2} + C. \]

Key Concepts

Trigonometric Integration

Trigonometric integration is a technique used when integrating functions involving trigonometric expressions. In this problem, the integral \( \int \frac{1}{\sqrt{1-x^2}} \, dx \) is a classic example of trigonometric integration that directly resembles the derivative of \( \sin^{-1}(x) \). Understanding these standard forms can help us solve integrals quickly.

• The function \( \frac{1}{\sqrt{1-x^2}} \) is directly connected to the inverse sine function.
• Recognizing these forms saves time and allows us to bypass complex calculations.

To approach these problems effectively, memorize the derivatives of inverse trigonometric functions. For instance, \( \frac{d}{dx} [\sin^{-1}(x)] = \frac{1}{\sqrt{1-x^2}} \). This knowledge effortlessly guides us to the integral's solution as \( \sin^{-1}(x) + C \), with \( C \) representing the constant of integration. By using trigonometric integration, we transform complicated expressions into familiar and manageable forms.

Substitution Method

The substitution method is a powerful tool in solving integrals. It simplifies the integration process by transforming a difficult integral into an easier one. In this exercise, we applied substitution to the integral \( \int \frac{x}{\sqrt{1-x^2}} \, dx \). This is how it works:

• Select an expression within the integral to substitute, such as \( u = 1 - x^2 \).
• Compute the derivative, \( du = -2x \, dx \), and rearrange it to replace \( x \, dx \) with \(-\frac{1}{2} \, du\).
• Substitute \( u \) and \( du \) into the integral, transforming it into \( \frac{1}{2} \int \frac{1}{\sqrt{u}} \, du \).

This substitution turns the original integral into a more straightforward form. The solution involves integrating \( u^{-\frac{1}{2}} \), resulting in \( 2\sqrt{u} + C \). Finally, revert the substitution by replacing \( u \) back with \( 1 - x^2 \) to complete the solution. Always remember to adjust for any constants introduced during substitution.

Fractional Integration

Fractional integration involves splitting a complex fraction into simpler parts, making the integration process easier. In this exercise, \( \frac{1-x}{\sqrt{1-x^2}} \) is separated into two smaller fractions.

• Decompose the integral: \( \int \frac{1-x}{\sqrt{1-x^2}} \, dx = \int \frac{1}{\sqrt{1-x^2}} \, dx - \int \frac{x}{\sqrt{1-x^2}} \, dx \).
• This separation allows each fraction to be tackled independently, often employing different integration techniques.

By splitting the fraction, each segment becomes identifiable with simpler, standard forms or becomes suited for substitution. This approach demonstrates how fractional integration provides a strategic way to manage integrals, leading to the successful solution of seemingly complex expressions.