Problem 17 - Calculus of a Single Variable

Question

Finding an Indefinite Integral In Exercises \(1-20\) , find the indefinite integral. $$ \int \frac{x-3}{x^{2}+1} d x $$

Step-by-Step Solution

Answer

The integral is equal to \(0.5ln(x^{2} + 1) - 3 \arctan(x) + C\), where C is the constant of integration.

Step 1: Decouple the fraction

Rewrite \(\int \frac{x-3}{x^{2}+1} dx\) as \(\int \frac{x}{x^{2}+1} dx - \int \frac{3}{x^{2}+1} dx\)

Step 2: Apply integration techniques

For the first integral, use a substitution method, set \(u = x^{2} + 1, du = 2xdx\). This transforms the first integral into \(0.5\int\frac{du}{u}\), which is easy to solve. For the second integral, recognize that the integral is in the form of \(\int \frac{1}{x^{2}+a^{2}} dx\), which is \(\frac{1}{a} \arctan(\frac{x}{a})\). So integrate to obtain \(3 \arctan(x)\).

Step 3: Combine the results

Add the results of the two integrals to get the final result, so we have \(0.5ln|x^{2} + 1| - 3 \arctan(x)\).

Key Concepts

Substitution Method

The substitution method is a popular technique used in integration to simplify complex integrals. It involves replacing part of the integral with a new variable to make the integration process easier. In this method, we identify a part of the expression that complicates the integration, often an inner function. We set this part equal to a new variable, typically labeled as "u," and differentiate it to find the corresponding derivative.

• Identify the substitution: Look for an expression inside the integral that can be simplified by substitution.
• Differentiate: Compute the derivative of your substitution to replace the variables of integration.
• Transform and Integrate: Rewrite the integral in terms of the new variable and solve it, often resulting in simpler integrals.

For example, in the problem \(\int \frac{x}{x^{2}+1} dx\), we use \(u = x^{2} + 1\), which transforms the integral into a simpler form, \(0.5 \int \frac{du}{u}\). This substitution makes the integral much easier to solve.

Integration Techniques

Integration techniques refer to various strategies and methodologies used to find integrals more easily. Different integrals require different techniques based on their structure.

• Substitution: Used when there is a nested function within an integral, often simplifying it into a basic integrable form.
• Partial Fractions: Useful for rational functions where the numerator's degree is less than the denominator's.
• Trigonometric Integration: Employed when dealing with integrals involving trigonometric functions.

In our problem, the integral was split into two parts, and each part applied different techniques. Substitution made \(\int \frac{x}{x^{2}+1} dx\) more straightforward, while recognizing the arctangent form helped solve \(\int \frac{3}{x^{2}+1} dx\). These methods simplify calculations and enhance our ability to tackle complex integral problems.

Arctan Integration

Arctan integration is useful when the integral is in the form \(\int \frac{1}{x^{2}+a^{2}} dx\), which corresponds to the antiderivative of the arctangent function. The solution follows a standard formula.

• Recognize the form: Identify if the integrand matches \(\frac{1}{x^2 + a^2}\). If yes, it can be integrated using arctan.
• Apply the formula: The integral becomes \(\frac{1}{a} \arctan\left(\frac{x}{a}\right) + C\), where \(a\) is a constant and \(C\) is the constant of integration.

In our exercise, the integral \(\int \frac{3}{x^{2}+1} dx\) was recognized as fitting this form with \(a = 1\), resulting in \(3 \arctan(x)\). This integration approach helps solve integrals involving sums of squares without complex manipulation.

Natural Logarithm Integration

Natural logarithm integration often arises when dealing with integrals of the form \(\int \frac{du}{u}\). This is because the antiderivative of \(\frac{1}{u}\) is the natural logarithm \(\ln|u| + C\).

• Recognize the form: See if substitution or manipulation reduces the integral to \(\frac{1}{u}\).
• Integrate using the logarithm: The resulting integral is \(\ln|u| + C\).

In this problem, after substitution \(u = x^{2} + 1\), we derived \(0.5 \int \frac{du}{u}\), leading to the result \(0.5 \ln|x^{2} + 1|\). This simple logarithmic form makes certain integrals straightforward to compute using basic rules, emphasizing the importance of recognizing and applying the appropriate integration technique.