Question

Use a table of integrals to evaluate the following integrals. $$ \int \frac{d y}{\sqrt{4-y^{2}}} $$

Short answer

The integral evaluates to \( \arcsin\left(\frac{y}{2}\right) + C \).

Step-by-step solution

Recognize the Integral Form

Recognize that the given integral \( \int \frac{d y}{\sqrt{4-y^{2}}} \) matches an entry in the standard table of integrals that includes integrals of the form \( \int \frac{d x}{\sqrt{a^2 - x^2}} = \arcsin\left(\frac{x}{a}\right) + C \). In our case, \( x = y \) and \( a = 2 \).

Apply the Formula

Substitute \( a = 2 \) and \( x = y \) into the formula to solve the integral. The resulting expression for the integral is \( \arcsin\left(\frac{y}{2}\right) + C \) where \( C \) is the constant of integration.

Key concepts

Integration

Integration is a central concept in calculus, often described as the reverse process of differentiation. It entails finding a function whose derivative matches a given function. This is called the 'antiderivative' of the function. Understanding integration is crucial because it allows us to solve problems involving areas, volumes, and even physics-related problems such as finding the work done by a variable force.

• Definite Integrals: These involve limits of integration and yield a specific number representing the accrued quantity over an interval.
• Indefinite Integrals: These do not have specific limits and represent a family of functions. The solution includes a constant of integration, denoted as "C".

In our exercise, we're dealing with an indefinite integral. Recognizing the pattern of the integral and matching it with a known formula is a great strategy for solving these problems efficiently.

Trigonometric Integrals

When tackling integrals involving trigonometric functions or requiring trigonometric substitutions, we are dealing with trigonometric integrals. These often include patterns where functions like sine, cosine, or variations thereof are involved. In many cases, familiarizing yourself with standard identities and integrals can ease the process significantly.
For instance, in our problem, noticing the integral form \( \int \frac{d y}{\sqrt{4-y^{2}}} \) hints that a trigonometric relationship or identity might simplify the process. The structure of \( \sqrt{4-y^{2}} \) suggests a connection to the identity \( \sin^2\theta + \cos^2\theta = 1 \). This clue can help transition to a trigonometric function in the solution. Here, using the arcsine function provides the answer, which relates closely to the Pythagorean identity.

Standard Integral Formulas

Standard integral formulas are pre-determined solutions for specific types of integrals, readily available to help quickly solve problems. For students, these are like a toolbox where each formula addresses a particular kind of expression. They are invaluable when working with complex integrals that match known forms.

• Example: The standard integral \( \int \frac{d x}{\sqrt{a^2 - x^2}} = \arcsin\left(\frac{x}{a}\right) + C \)
• Tools like integral tables or algebra software can offer these formulas for swift reference.

In exercises like ours, recognizing that the integral fits one of these standard forms allows us to write down the solution immediately. The familiarity with these formulas saves time and simplifies calculations, allowing us to focus on understanding the underlying concepts rather than repetitive computation.