Problem 34 - Calculus Single Variable

Question

Evaluate the given integral. $$ \int \frac{x^{2}+2}{x^{2}+1} d x $$

Step-by-Step Solution

Answer

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

Step 1: Simplify the Integrand

The integrand \( \frac{x^2+2}{x^2+1} \) can be split into two separate fractions: \( 1 + \frac{1}{x^2+1} \). This is because \( x^2 + 2 = (x^2 + 1) + 1 \). So, rewrite the integral as \( \int \left( 1 + \frac{1}{x^2+1} \right) dx \).

Step 2: Integrate the First Part

The integral of \( 1 \) with respect to \( x \) is \( x \). So, \( \int 1 \, dx = x \).

Step 3: Integrate the Second Part

The integral of \( \frac{1}{x^2+1} \) with respect to \( x \) is \( \tan^{-1}(x) \), since the derivative of \( \tan^{-1}(x) \) is \( \frac{1}{x^2+1} \). Therefore, \( \int \frac{1}{x^2+1} \, dx = \tan^{-1}(x) \).

Step 4: Combine the Results

Combine the results from Step 2 and Step 3 to find the integral: \( \int \left( 1 + \frac{1}{x^2+1} \right) dx = x + \tan^{-1}(x) + C \), where \( C \) is the constant of integration.

Key Concepts

Integrand Simplification

In calculus, simplifying the integrand is often the first step in evaluating an integral. The integrand is the function inside the integral that needs to be integrated. Here, the original integrand is \( \frac{x^2+2}{x^2+1} \). To make things simpler, we can express this as two separate fractions by recognizing that \( x^2 + 2 = (x^2 + 1) + 1 \). This means:

• \( \frac{x^2 + 2}{x^2 + 1} = 1 + \frac{1}{x^2 + 1} \)

This simplification makes integration easier, as we can now handle each term separately.

• The first term, \( 1 \), becomes straightforward to integrate.
• The second term, \( \frac{1}{x^2+1} \), can be recognized as a standard form that relates to the arctangent function.

Breaking down a complex expression like this helps in solving the integral more efficiently.

Trigonometric Substitution

Trigonometric substitution is a technique often used in integral calculus when the integrand contains expressions like \( \sqrt{a^2 - x^2} \), \( \sqrt{a^2 + x^2} \), or \( \sqrt{x^2 - a^2} \). It transforms the integrand into a form that is easier to handle using trigonometric identities.
In our problem, while we did not perform traditional trigonometric substitution, we took advantage of recognizing the integral form \( \frac{1}{x^2+1} \), which is the derivative of \( \tan^{-1}(x) \). With some practice, you can often spot such trigonometric forms, making integration a breeze.

• Always look for patterns that fit known integrals involving trigonometric functions.
• This technique simplifies the computation and often leads directly to a solution you can recognize from standard tables of integrals.

Constant of Integration

In indefinite integration, after finding the antiderivative of a function, you must add a constant of integration, denoted as \( C \). This constant accounts for any arbitrary constant that could have been present when differentiating the original function.
The constant of integration is crucial because:

• Indefinite integrals represent a family of functions, and the constant ensures all possible solutions are represented.
• Without \( C \), you only represent a single function, potentially missing other valid solutions.

In our example, after integrating the simplified form of the integrand, the complete solution is given by \( x + \tan^{-1}(x) + C \). Remembering \( C \) is essential every time you perform indefinite integration.