Problem 2 - Thomas Calculus

Question

The integrals in Exercises \(1-44\) are in no particular order. Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate. When necessary, 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 is \( x - \arctan(x) + C \).

Step 1: Identify the Integration Technique

In this exercise, the integrand is \( \frac{x^2}{x^2+1} \). A common approach here is to split the fraction in such a way that makes it easier to integrate. To do so, we can rewrite \( \frac{x^2}{x^2+1} \) as something composed of simpler terms.

Step 2: Decompose the Integrand

Rewrite \( \frac{x^2}{x^2+1} \) as \( 1 - \frac{1}{x^2+1} \). This is equivalent because \( \frac{x^2}{x^2+1} = 1 - \frac{1}{x^2+1} \), derived from the identity \( \frac{x^2 + 1 - 1}{x^2 + 1} \).

Step 3: Integrate Each Term Separately

Now, integrate the expression. We have: \[ \int \left(1 - \frac{1}{x^2+1}\right) dx = \int 1 \, dx - \int \frac{1}{x^2+1} \, dx \].The first integral, \( \int 1 \, dx \), is simply \( x \).The second integral, \( \int \frac{1}{x^2+1} \, dx \), is a standard form and its integral is \( \arctan(x) \).

Step 4: Write the Final Solution

Combine the results from the previous step: \[ x - \arctan(x) + C \]where \( C \) is the constant of integration.

Key Concepts

Integration by Decomposition

Integration by decomposition is a technique where you break down a complex integrand into simpler parts to make integration easier. This method allows you to reframe a difficult integral into a form that you can manage with straightforward integration techniques. In the given exercise, the integrand is \[\frac{x^2}{x^2+1}\]. To apply decomposition, notice that you can rewrite it as \[1 - \frac{1}{x^2+1}\]. This insight comes from realizing that \[\frac{x^2 + 1 - 1}{x^2 + 1} = \frac{x^2 + 1}{x^2 + 1} - \frac{1}{x^2 + 1} = 1 - \frac{1}{x^2 + 1}\]. This simple yet powerful step transforms a challenging integral into two manageable ones:

• \( \int 1 \, dx \)
• \( \int \frac{1}{x^2+1} \, dx \)

By decomposing the original integrand, we have simplified the problem significantly. Each part can now be integrated with ease, utilizing standard integrals and identities.

Standard Integrals

Standard integrals are basic integral formulas that you should know by heart, which simplify the integration process. They are fundamental building blocks for evaluating more complex integrals. Recognizing when these standard formulas apply within a decomposed integral can save a lot of time and effort. In our exercise, after decomposition, we are left with two integrals to solve:

• \( \int 1 \, dx \)
• \( \int \frac{1}{x^2+1} \, dx \)

The integral \( \int 1 \, dx \) is straightforward. Its result is simply the variable of integration, \( x \), as it represents the accumulation of a constant, 1, over variable \( x \). Similarly, \( \int \frac{1}{x^2+1} \, dx \) is a classic integral that yields \( \arctan(x) \). Knowing these standard forms allows you to swiftly arrive at the solution for the overall integral, which combines these results as:\[x - \arctan(x) + C\]. Where \( C \) represents the constant of integration, capturing the family of all possible antiderivatives.

Trigonometric Identities

Trigonometric identities are valuable tools in integration, particularly when dealing with integrals involving trigonometric functions or when they can be transformed into such. Although in this specific exercise, trigonometric identities do not directly rewrite the given integral, they enable the evaluation of \[\int \frac{1}{x^2+1} \, dx\]. This integral is solved using the identity that connects it to the arctangent function:\[\int \frac{1}{x^2 + 1} \, dx = \arctan(x) + C\]. Remembering this relationship is crucial because it allows you to recognize parts of an integral that simplify directly into an inverse trigonometric function. Trigonometric identities like this one often show up in calculus, so being familiar with them can significantly streamline your problem-solving process. By practicing and memorizing these key identities, you can handle a variety of integration problems with much greater ease. This adds invaluable efficiency and effectiveness to your mathematical toolkit.