Problem 33 - Calculus Single Variable

Question

Evaluate the given integral. $$ \int \frac{2 x+3}{x^{2}+1} d x $$

Step-by-Step Solution

Answer

\( \ln|x^2+1| + 3 \cdot \tan^{-1}(x) + C \)

Step 1: Recognize the Integral Form

The given integral is \( \int \frac{2x+3}{x^2+1} \, dx \). This fraction is suited to be split for easier integration. Identify the components: \( 2x \) and \( 3 \) in the numerator and divide them over the denominator \( x^2+1 \).

Step 2: Separate the Fraction

Rewrite the integral by separating the terms of the numerator: \[ \int \frac{2x}{x^2+1} \, dx + \int \frac{3}{x^2+1} \, dx. \] Now, we have two simpler integrals to handle.

Step 3: Solve the First Integral

For \( \int \frac{2x}{x^2+1} \, dx \), notice that the numerator \( 2x \) is the derivative of the denominator \( x^2+1 \). Thus, this integral can be evaluated using the natural logarithm rule:\[ \int \frac{2x}{x^2+1} \, dx = \ln|x^2+1| + C_1. \]

Step 4: Solve the Second Integral

For \( \int \frac{3}{x^2+1} \, dx \), recognize that the integral is a standard form which results in an arctangent function. So:\[ \int \frac{3}{x^2+1} \, dx = 3 \cdot \tan^{-1}(x) + C_2. \]

Step 5: Combine Results and Simplify

Combine the results of the two integrals into a single expression:\[ \int \frac{2x+3}{x^2+1} \, dx = \ln|x^2+1| + 3 \cdot \tan^{-1}(x) + C. \]Here, \( C \) represents the combined constant of integration \( C_1 + C_2 \).

Key Concepts

Natural Logarithm Integration

When solving integrals, especially those from fractions, recognizing the natural logarithm is a helpful approach. This happens when the numerator is the derivative of the denominator. For example, in the integral \( \int \frac{2x}{x^2+1} \, dx \), the numerator \( 2x \) is exactly the derivative of the denominator \( x^2+1 \). Therefore, this integral follows the natural log rule.To integrate, it simplifies to the natural logarithm of the absolute value of the denominator:

• \( \int \frac{2x}{x^2+1} \, dx = \ln|x^2+1| + C \),

where \( C \) is the constant of integration. Breaking up fractions in this way can clarify complex integrals, guiding you to a solution through familiar logarithmic properties.

Arctangent Integration

The arctangent function is a key player when integrating fractions with a denominator in the form \( x^2+1 \). Recognizing these patterns can turn a seemingly difficult integral into a much simpler form. For the integral \( \int \frac{3}{x^2+1} \, dx \), the form matches the derivative of the arctangent function: \( \tan^{-1}(x) \).Understanding this relation allows you to integrate easily:

• \( \int \frac{3}{x^2+1} \, dx = 3 \cdot \tan^{-1}(x) + C \),

where the factor of 3 simply scales the result. This pattern comes up often, and recognizing it early can save time and effort in determining the result of an integral. Remember, mastery of these standard forms can make tackling integrals a much smoother process.

Constant of Integration

The constant of integration, denoted as \( C \), appears in indefinite integrals as a mode of expressing the family of all possible antiderivatives. Since integration is the reverse process of differentiation, when you differentiate a constant, it disappears. Therefore, any constant added to a function being integrated would also disappear upon differentiation.When dealing with integrals that have been broken down, such as:

• \( \ln|x^2+1| + 3 \cdot \tan^{-1}(x) + C \)

this constant \( C \) is the cumulative representation of potential constants from both parts of split integrals. Whether you're dealing with simple or more complex functions, never forget to include \( C \). It signifies completeness in your solution, covering all possibilities of the integrated function.