Problem 76 - Calculus for Biology and Medicine

Question

Compute the indefinite integrals. $$ \int\left(1-\frac{x^{2}}{1+x^{2}}\right) d x $$

Step-by-Step Solution

Answer

The indefinite integral is \(\arctan(x) + C\).

Step 1: Simplify the Integrand

The integrand is \(1 - \frac{x^2}{1+x^2}\). Start by rewriting the fraction \(\frac{x^2}{1+x^2}\) in a simplified form. Notice that it can be rewritten using the identity: \(\frac{x^2}{1+x^2} = 1 - \frac{1}{1+x^2}\). Thus, the integrand becomes:\[\int \left( 1 - \left(1 - \frac{1}{1+x^2} \right) \right) dx = \int \frac{1}{1+x^2} dx.\]

Step 2: Recognize the Standard Integral

The simplified integrand \(\frac{1}{1+x^2}\) is a standard integral that can be directly integrated. Recall the standard integral:\[\int \frac{1}{1+x^2} dx = \arctan(x) + C,\]where \(C\) is the constant of integration.

Key Concepts

Integrand Simplification

Simplifying the integrand is often the first step in solving an indefinite integral. In this problem, start by looking at the expression inside the integral:\[1 - \frac{x^2}{1+x^2}\]To simplify, notice that you can manipulate the fraction \( \frac{x^2}{1+x^2} \) using an identity to express it differently. By recognizing that:\[\frac{x^2}{1+x^2} = 1 - \frac{1}{1+x^2}\]you can substitute this into the integrand. This simplifies the integrand to:\[1 - \left(1 - \frac{1}{1+x^2}\right) = \frac{1}{1+x^2}\]With this process, we have transformed the integrand into a more familiar form, which is crucial for the next step in integration. The simplification helps identify standard integrals, making it easier to solve.

Standard Integral Recognition

Recognizing standard integrals is a key strategy in finding antiderivatives. Often, complex integrands can be transformed into recognizable forms. Through simplification, the integrand in this exercise becomes:\[\frac{1}{1+x^2}\]This expression is a well-known standard integral. Students should remember that:\[\int \frac{1}{1+x^2} \, dx = \arctan(x) + C\]where \( C \) is the constant of integration.Recognizing this pattern allows us to immediately find the antiderivative without additional steps. Memorizing various standard integrals can save significant time and simplify the process of integration. It's very efficient when you encounter typical forms like this one.

Arctan Integration

Understanding the integration involving the arctangent function is a handy tool. The function \( \arctan(x) \), or the inverse tangent function, is tied directly to the integral:\[\int \frac{1}{1+x^2} \, dx = \arctan(x) + C\]where \( C \) signifies an arbitrary constant. This relationship stems from the derivative of the arctan function:\[\frac{d}{dx} \arctan(x) = \frac{1}{1+x^2}\]So, when integrating \( \frac{1}{1+x^2} \), we see it as reversing the differentiation of \( \arctan(x) \).Hence, any time you encounter such a form, you can confidently write the antiderivative as \( \arctan(x) + C \). Applying inverse trigonometric integrals, like this one, broadens the ability to handle diverse integration problems swiftly and effectively.