Problem 5 - Thomas Calculus

Question

The integrals in Exercises \(1-40\) are in no particular order. Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. $$ \int \frac{1-x}{\sqrt{1-x^{2}}} d x $$

Step-by-Step Solution

Answer

\(\arcsin x + \sqrt{1-x^2} + C\)

Step 1: Identify the Integral Type

The integral \( \int \frac{1-x}{\sqrt{1-x^{2}}} \, dx \) is a rational function divided by a square root. This suggests that a trigonometric substitution might simplify the expression.

Step 2: Choose a Substitution

For integrals involving \( \sqrt{1-x^2} \), a common substitution is \( x = \sin \theta \), because \( \sqrt{1-x^2} = \cos \theta \). Differentiating, we get \( dx = \cos \theta \, d\theta \). Substitute \( x = \sin \theta \) and \( dx = \cos \theta \, d\theta \) into the integral.

Step 3: Apply the Substitution

Substitute into the integral:\[\int \frac{1-x}{\sqrt{1-x^2}} \, dx = \int \frac{1-\sin \theta}{\cos \theta} \cdot \cos \theta \, d\theta = \int (1 - \sin \theta) \, d\theta.\]The expression simplifies nicely to \( \int (1 - \sin \theta) \, d\theta \).

Step 4: Integrate the Simplified Expression

Separate the integral into simpler parts:\[\int (1 - \sin \theta) \, d\theta = \int 1 \, d\theta - \int \sin \theta \, d\theta.\]These integrate to:\[\theta - (-\cos \theta) = \theta + \cos \theta + C.\]

Step 5: Back Substitute \( \theta \)

Recall that \( x = \sin \theta \) implies \( \theta = \arcsin x \). Substitute back:\[\theta + \cos \theta = \arcsin x + \sqrt{1-x^2}.\]Thus, the integral evaluates to:\[\int \frac{1-x}{\sqrt{1-x^2}} \, dx = \arcsin x + \sqrt{1-x^2} + C.\]

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a powerful technique used to simplify the integration of functions involving square roots. When you see integrals that contain expressions like \( \sqrt{1-x^2} \), a trigonometric substitution can often streamline the process.

Here's how it works:

• Recognize the form \( \sqrt{1-x^2} \). It hints at the trigonometric identity \( \sin^2 \theta + \cos^2 \theta = 1 \).
• Choose a substitution to match this identity. For \( \sqrt{1-x^2} \), substitute \( x = \sin \theta \), making \( \sqrt{1-x^2} = \cos \theta \).
• Convert \( dx \) appropriately: differentiate \( x = \sin \theta \) to get \( dx = \cos \theta \, d\theta \).

This substitution translates the integral into trigonometric terms, often simplifying it to a more standard form that's easier to handle. Remember, choosing the correct trigonometric substitution is crucial for transforming complex irrational functions into manageable expressions.

Integration Techniques

Integration techniques encompass a variety of strategies, all aimed at tackling a diverse set of integrative challenges. With trigonometric substitution, we've seen how substitution can simplify integrals. However, there are additional techniques that are handy for integrating various functions.

Common techniques include:

• Substitution Method: This involves changing variables to simplify the integral, frequently using a derivative to match parts of the original function.
• Integration by Parts: Useful for products of functions, this method uses the formula \( \int u \, dv = uv - \int v \, du \).
• Partial Fraction Decomposition: Often employed for rational functions, this technique breaks a fraction down into simpler fractions that are easier to integrate.

Each technique has its applications and choosing the right one depends on the form of the function you are integrating. Practice and familiarity with these methods enhance your ability to integrate complex expressions effectively.

Algebraic Methods

Algebraic methods in integration deal with manipulating algebraic expressions before performing the integration. The goal is to simplify the function to a form that is easier to integrate.

To employ algebraic methods, consider the following steps:

• Factorization: Look for ways to factor polynomials or algebraic expressions. This can sometimes simplify the integral considerably.
• Completing the Square: With quadratic expressions, completing the square can transform the integral into a more familiar shape.
• Simplification: Use algebraic identities to break down complex expressions. Simplifying before integrating can save you from much headache later on.

In the provided exercise, simplifying the integrand \( \frac{1-x}{\sqrt{1-x^2}} \) into something manageable was crucial. This method often pairs well with other integration techniques to tackle challenging integrals. Understanding and mastering these algebraic manipulations can significantly enhance your problem-solving toolkit in calculus.