Problem 2 - 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. $$ \int \frac{x^{2}}{x^{2}+1} d x $$

Step-by-Step Solution

Answer

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

Step 1: Analyzing the Integral

We need to evaluate the integral \( \int \frac{x^2}{x^2+1} \, dx \). It is a rational function where the numerator's degree is equal to the denominator's degree.

Step 2: Algebraic Manipulation

Divide the numerator by the denominator: \( \frac{x^2}{x^2+1} = 1 - \frac{1}{x^2+1} \). This simplifies the integral to two separate integrals: \( \int 1 \, dx - \int \frac{1}{x^2+1} \, dx \).

Step 3: Evaluating Each Part Separately

First, solve \( \int 1 \, dx = x + C_1 \), where \(C_1\) is the constant of integration. Next, solve \( \int \frac{1}{x^2+1} \, dx = \tan^{-1}(x) + C_2 \), where \(C_2\) is another constant of integration. Both results use standard integral formulas.

Step 4: Combining the Results

Combine the two results to express the solution of the original integral: \( x - \tan^{-1}(x) + C \). Here, \(C\) is the constant of integration that combines \(C_1\) and \(C_2\).

Key Concepts

Rational Functions

A rational function is a fraction where both the numerator and the denominator are polynomials. These types of functions often appear in calculus problems, particularly in integrals and derivatives. For our specific problem, the rational function is \( \frac{x^2}{x^2+1} \).

• The degree of the polynomial in the numerator is 2, which is the same as the degree of the polynomial in the denominator.
• This characteristic makes the function perfect for division because the degrees match up.

In the solution, we tackled this by dividing the expressions, resulting in a simpler form: \( 1 - \frac{1}{x^2+1} \). Therefore, much like simplifying fractions in arithmetic, dividing polynomials helps to break down the integral into more manageable pieces. This process is essential in solving complex rational functions in integration.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions like sine, cosine, and tangent. These identities hold true for all values of the variables where the functions are defined. In calculus, they can simplify integrals by transforming them into forms that are easier to evaluate.

• For instance, the identity \( \int \frac{1}{x^2+1} \, dx = \tan^{-1}(x) + C_2 \) is quite handy here.

This particular identity relates the inverse tangent (arctan) function to a simple rational function. Using it allows us to solve integrals more efficiently. When you see \( \frac{1}{x^2+1} \) in an integral, it’s a clear signal to use the inverse tangent function, effectively reducing the problem to a standard form. This transformation streamlines the calculus process and is a critical tool in your mathematical toolkit.

Substitution Method

The substitution method, also known as "u-substitution," is a technique used to simplify the process of integration by making a change of variables. It works by replacing a complex or awkward expression with a single variable.

• The goal is to transform the integral into a standard form, which is easier to solve.
• In our problem, however, the simplification steps made substitution unnecessary because the integral was already broken down into manageable pieces, \( 1 \) and \( \frac{1}{x^2+1} \).

However, understanding substitution is crucial for more complicated integrals where direct techniques are not applicable. Always be ready to identify parts of an integral that might benefit from substitution. It's an extremely potent method to have in your calculus arsenal, often making what seems daunting manageable through clever use of transformation.