Problem 13 - Calculus - AP Edition

Question

Evaluate the integral. $$\int \frac{d x}{\left(1-x^{2}\right)^{3 / 2}}$$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{x}{\sqrt{1-x^2}} + C \).

Step 1: Identify the Integral Type

The integral is of the form \( \int \frac{dx}{(1-x^2)^{3/2}} \). This is a trigonometric integral commonly associated with inverse trigonometric functions.

Step 2: Use the Trigonometric Substitution

Set \( x = \sin \theta \), then \( dx = \cos \theta \, d\theta \) and \( 1 - x^2 = \cos^2 \theta \). This substitution is helpful because it simplifies the square root in the denominator.

Step 3: Simplify the Integral

Substituting \( x = \sin \theta \) into the integral, we have: \[ \int \frac{\cos \theta \, d\theta}{(\cos^2 \theta)^{3/2}} = \int \frac{\cos \theta \, d\theta}{\cos^3 \theta} \] This simplifies to: \[ \int \frac{1}{\cos^2 \theta} \, d\theta = \int \sec^2 \theta \, d\theta \]

Step 4: Integrate with Respect to \(\theta\)

The integral \( \int \sec^2 \theta \, d\theta \) is a standard integral with the antiderivative \( \tan \theta + C \).

Step 5: Back-Substitute for \(x\)

Since \( x = \sin \theta \), then \( \theta = \arcsin x \). Therefore, \( \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{x}{\sqrt{1-x^2}} \). Substitute this back into our result: \[ \tan \theta + C = \frac{x}{\sqrt{1-x^2}} + C \]

Step 6: Final Answer

The integral evaluates to: \[ \int \frac{dx}{(1-x^2)^{3/2}} = \frac{x}{\sqrt{1-x^2}} + C \]

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are used to find angles given trigonometric values. They are essentially the opposite of the usual sine, cosine, and tangent functions, and they are represented as arcsine (\(\arcsin\)), arccosine (\(\arccos\)), and arctangent (\(\arctan\)). These functions are essential when we perform integration techniques involving trigonometric substitutions, where they help in reversing the substitution process.

In this exercise, the substitution \(x = \sin\theta\) allows us to use inverse trigonometric functions to revert back from \(\theta\) to \(x\) after integrating. As a result, \(\theta = \arcsin x\) becomes crucial. Understanding how inverse trigonometric functions interact with regular trigonometric functions is key to properly manipulating and solving integrals involving these substitutions.

Integration Techniques

Integration techniques involve various methods to solve integrals, a crucial part of calculus. One common technique is substitution, which simplifies the integral by changing variables. When dealing with trigonometric integrals, trigonometric substitution is often useful.

The process involves:

• Identifying a trigonometric identity that simplifies the integral.
• Choosing a substitution, such as \(x =\sin\theta\), which helps reduce complex expressions.
• Simplifying the integral using this substitution.

In the given solution, the substitution of \(x = \sin\theta\) transforms the denominator from \((1-x^2)^{3/2}\) to \(\cos^3 \theta\), allowing the integral of \(\sec^2 \theta\) to be straightforwardly computed. Mastering these techniques is invaluable for solving diverse integral problems.

Definite and Indefinite Integrals

Integrals can be classified into definite and indefinite integrals. An indefinite integral, as in this exercise, is represented without upper and lower limits and includes a constant of integration \(C\). It represents a family of functions and is the reverse process of differentiation.

On the other hand, definite integrals are evaluated over a specific interval, yielding a numerical value. They do not include the constant \(C\). Understanding these differences is crucial when applying integration techniques to solve problems.

In our case, since the integral \(\int \frac{dx}{(1-x^2)^{3/2}}\) is indefinite, the final solution must include \(C\). This additional term accounts for any constant value that could have been lost during differentiation. Recognizing when to use definite versus indefinite techniques helps streamline the solving process.