Question

For the following exercises, find the antiderivatives for the functions.\(\int-\frac{d x}{x \sqrt{1-x^{2}}}\)

Short answer

The antiderivative is: \(-\ln|x| \arcsin(x) + C\).

Step-by-step solution

Simplify the Function

The given function is \(-\frac{1}{x \sqrt{1-x^{2}}}\). Let's simplify it to understand which integration technique to use. \(-\frac{1}{x}\) is a constant multiple outside the main expression \(\frac{1}{\sqrt{1-x^{2}}}\).

Identify the Integration Method

Recognize that \( \frac{1}{\sqrt{1-x^{2}}} \) is the derivative of \( \arcsin(x) \). This suggests the use of the antiderivative formula for \( \arcsin(x) \), which is \( \int \frac{1}{\sqrt{1-x^{2}}} \ dx = \arcsin(x) + C \).

Incorporate Constant Multiple

Now, integrate \(-\frac{1}{x}\) separately. Multiplication of integrals is not directly applicable here, so observe that the integral becomes a mere constant multiplier. Thus, our integral transforms to the antiderivative problem of \(-\int \frac{dx}{x}\).

Compute Antiderivative

The antiderivative for \(-\int \frac{dx}{x}\) is \(-\ln|x| + C\). Here, \(C\) is the integration constant. Therefore, the overall solution to our original function thus becomes \(-\ln|x| \arcsin(x) + C\), combining the two parts.

Key concepts

Integration Techniques

Integration techniques are methods used to find antiderivatives, also known as indefinite integrals. Understanding which technique to use depends on the form of the function we are dealing with. Let's explore some common techniques that are particularly useful:

• **Basic Antidifferentiation**: This is about directly applying known antiderivative formulas, much like reversing differentiation. For instance, the antiderivative of a power function can be found using the reverse of the power rule, except for the special case when the power is -1.
• **Substitution**: When the integrand is a composite function, substitution can simplify the process. It involves changing variables to make the integration easier.
• **Integration by Parts**: This technique comes in handy when dealing with the product of two functions. The rule is derived from the product rule for differentiation and is particularly useful for functions like polynomials multiplied by transcendental functions.
• **Trigonometric Integrals**: Useful when dealing with products or powers of trigonometric functions, these often involve leveraging trigonometric identities.

In the exercise at hand, the key integration method is recognizing specific derivatives—in this case, the derivative of arcsine, which transforms the problem into a format we can readily integrate.

Arcsin Function

The arcsine function, denoted as \( \arcsin(x) \), is the inverse of the sine function over its principal range \([-\frac{\pi}{2}, \frac{\pi}{2}]\). It takes a value from these sine's outputs and returns their corresponding angle.

• **Derivative of Arcsine**: The derivative of the arcsine function can be represented as \( \frac{d}{dx}[\arcsin(x)] = \frac{1}{\sqrt{1-x^2}} \). This is essential in identifying integrals involving arcsine.
• **Antiderivative Relation**: When identifying integrals, recognizing \( \frac{1}{\sqrt{1-x^2}} \) as the derivative of \( \arcsin(x) \) allows us to directly find its antiderivative as \( \arcsin(x) + C \) where \( C \) is a constant.

The arcsin function helps in tackling problems where the variable inside the integral follows a specific form, typically associated with a resulting in the derivative structure seen in the arcsine. Understanding this connection simplifies integration tasks involving these forms.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a logarithm to the base \(e\), where \(e\) is Euler's number (approximately 2.71828). In integration, particularly for the exercise we are focusing on, understanding the derivative and antiderivative of the logarithmic function is crucial.

• **Derivative of Natural Logarithm**: The derivative is \( \frac{d}{dx}[\ln|x|] = \frac{1}{x} \). This inverse relationship implies that the antiderivative of \( \frac{1}{x} \) is \( \ln|x| \).
• **Importance in Integration**: Recognizing \( \frac{1}{x} \) allows the calculation for an antiderivative to yield \( \ln|x| + C \). This encourages the use of logarithmic properties in integration to simplify expressions, especially when dealing with simpler rational functions.

In our specific example, the negative multiplication leading to \(-\ln|x|\) highlights a nuanced approach. It demonstrates how constants factor into the use of logarithms, with subtraction invoked by the negative multiplier, completing our antiderivative computation.