Problem 1 - Thomas Calculus

Question

Use the table of integrals at the back of the book to evaluate the integrals. \(\int \frac{d x}{x \sqrt{x-3}}\)

Step-by-Step Solution

Answer

\( \int \frac{d x}{x \sqrt{x-3}} = \frac{2}{3} \text{arctan} \left( \sqrt{\frac{x-3}{3}} \right) + C \).

Step 1: Identify the Integral Form

Look at the integral \(\int \frac{d x}{x \sqrt{x-3}}\). Our goal is to match it with one of the standard forms from the table of integrals. A common form in these tables is \( \int \frac{1}{x\sqrt{x-a}} \, dx \), which matches our integral if \(a = 3\).

Step 2: Locate the Standard Formula

Find the standard formula for \( \int \frac{1}{x\sqrt{x-a}} \, dx \) in the table of integrals. The corresponding formula is usually, \( \frac{2}{a} \text{arctan} \Bigg( \sqrt{\frac{x-a}{a}} \Bigg) + C \), where \( a \) is a constant in the expression.

Step 3: Substitute and Evaluate

Use the identified standard formula and substitute \( a = 3 \). This gives us the evaluated integral: \( \frac{2}{3} \text{arctan} \Bigg( \sqrt{\frac{x-3}{3}} \Bigg) + C \).

Key Concepts

Table of Integrals

When it comes to solving integrals in calculus, a table of integrals can be highly beneficial. This table lists various integrals of common functions. This means you can find an integral in the table and use it to solve problems without computing them from scratch every time. Tables of integrals often include integrals of polynomial functions, exponential functions, trigonometric functions, and more complex forms.

• They are especially useful for students and professionals who need quick access to integral solutions.
• In the context of the exercise, identifying the correct form within the table is the first step.
• This saves time and reduces the possibility of making calculation errors.

The integral \( \int \frac{d x}{x \sqrt{x-3}} \), as discussed in our example, matches a form you might see in such a table. These tables simplify the decision-making process by listing various "standard" forms that you can match to your problem.

Definite Integral

Definite integrals are a key concept in calculus providing the signed area under a curve over a certain interval. Unlike indefinite integrals, which give a family of functions, definite integrals have limits that are specific points on the x-axis.The expression for a definite integral is given by \( \int_{a}^{b} f(x) \, dx \), where \(a\) and \(b\) represent the lower and upper bounds respectively.

• Calculating a definite integral involves finding the antiderivative of the function and evaluating it at the upper and lower limits.
• These calculations are useful in determining areas, solving real-world problems, and more.

While our original problem was an indefinite integral, transitioning to definite integrals would involve applying these boundaries to our result, further showcasing how integrals can be transformed using these limits.

Standard Integral Forms

Standard integral forms are pre-established formulas that are widely recognized and used in integral calculus. These forms simplify many integration problems by providing ready-made solutions for common integrals.For instance, in the problem we discussed, the form was \( \int \frac{1}{x\sqrt{x-a}} \, dx \). This was identified and matched in the table of integrals.

• Using these standard forms reduces the complexity of deriving integrals on your own.
• Once you identify a matching standard form, you can use it directly by substituting the variables or constants as needed, like substituting \(a = 3\).

By making use of these standardized forms, calculus becomes more accessible and less prone to errors, particularly with more complex expressions. Understanding and recognizing these forms is a crucial skill in integral calculus.