Question

Use a table of integrals to evaluate the following integrals. $$ \int \frac{x+3}{x^{2}+2 x+2} d x $$

Short answer

\( \int \frac{x+3}{x^2+2x+2} \, dx = \frac{1}{2} \ln|x^2+2x+2| + 2 \arctan(x+1) + C \).

Step-by-step solution

Identify Form of Integral

Examine the integral \( \int \frac{x+3}{x^{2}+2x+2} \, dx \) to identify its form. Notice that the integral has a rational function where the numerator \( (x+3) \) is a polynomial of degree less than the denominator \( (x^2 + 2x + 2) \). This suggests a potential use of the integral form for rational expressions or decomposition using a table of integrals.

Perform Partial Fraction Decomposition

Observe the denominator \( x^2 + 2x + 2 \). It cannot be factored into linear terms with real coefficients, as the discriminant \( (2^2 - 4 \times 1 \times 2) = -4 \) is negative, indicating complex roots. We use decomposition where the numerator is split into two separate terms so that it conforms to solutions in integral tables: \( \frac{x+3}{x^2+2x+2} = \frac{A \cdot \frac{d}{dx}(x^2+2x+2) + B}{x^2 + 2x + 2} \).

Simplify using Derivatives

The derivative of the denominator \( x^2+2x+2 \) is \( 2x + 2 \). We express the numerator in terms of this derivative: \( x + 3 = A \, (2x + 2) + B \). Expanding gives \( 2Ax + 2A + B = x + 3 \). Set coefficients equal: \( 2A = 1 \) and \( 2A + B = 3 \). Solve for \( A \) and \( B \).

Solve Equations for Coefficients

From \( 2A = 1 \), we find \( A = \frac{1}{2} \). Substitute into \( 2A + B = 3 \) to get \( 1 + B = 3 \), leading to \( B = 2 \).

Substitute and Integrate

Replace in the decomposed form and integrate: \( \int \frac{x+3}{x^2 + 2x + 2} \, dx = \frac{1}{2} \int \frac{2x + 2}{x^2 + 2x + 2} \, dx + 2 \int \frac{1}{x^2 + 2x + 2} \, dx \). The first integral is natural log \( \frac{1}{2} \ln|x^2 + 2x + 2| \). Completing the square for the second part gives \( x^2 + 2x + 2 = (x+1)^2 + 1 \), leading to \( 2 \times \text{arctan}(x+1) \).

Write Final Integral Result

Combine results from the integrals to obtain the final answer: \[ \int \frac{x+3}{x^2+2x+2} \, dx = \frac{1}{2} \ln|x^2 + 2x + 2| + 2 \arctan(x+1) + C \]. Here, \( C \) represents the constant of integration.

Key concepts

Rational Functions

Rational functions are an essential concept in calculus, particularly in integral calculus when dealing with integration. A rational function is essentially a quotient where both the numerator and the denominator are polynomials. For example, in \( \frac{x+3}{x^{2}+2x+2} \), the numerator and denominator are polynomials of degree 1 and 2, respectively.

Students often need to deal with rational functions when performing integrals, since they can appear in complex forms requiring decomposition and other steps for simplification.

• If the degree of the numerator is lower than the denominator, this often indicates that partial fraction decomposition might be useful for simplification.
• If it is higher or equal, polynomial long division might be necessary before applying other techniques.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down complex rational expressions into simpler, more manageable parts. This proves extremely helpful while resolving integrals involving rational functions.

Approaches like decomposition can transform \( \frac{x+3}{x^{2}+2x+2} \)into separate components which are easier to integrate using standard integral formulas. This involves identifying irreducible quadratic factors that exist in the denominator and assigning respective numerators that allow for integration:

• The discriminant of the quadratic tells us if the factor can be further simplified (factored).
• Create decomposition terms and equate to find constants that satisfy the initial expression.

Understanding and applying this method is crucial in integral calculus, especially when dealing with complex rational functions.

Definite and Indefinite Integrals

In integral calculus, definite and indefinite integrals are fundamental concepts. While the original problem does not provide limits of integration, implying it deals with an indefinite integral,
understanding both types is important. An indefinite integral leads to a general form plus a constant \( C \), while definite integrals calculate the area under a curve, providing numerical answers.

The indefinite integral \( \int \frac{x+3}{x^{2}+2x+2} \, dx \) resulted in:

• \( \frac{1}{2} \ln|x^{2}+2x+2| \)
• \(2 \arctan(x+1)\)
• \(+ \ C \)

Use definite integrals when solving for exact areas between limits, taking into account the specific interval on the x-axis where you evaluate the function.

Natural Logarithm Integration

Integration of functions involving logarithms, especially natural logarithms indicated by \( \ln \), often occurs with rational functions. The natural logarithm function, \( \ln(x) \), is defined as the integral of \( \frac{1}{t} \) from 1 to \( x \).

In the provided exercise, we observe natural logarithm integration as part of the solution:

• Within the expression \( \frac{1}{2} \ln|x^{2}+2x+2| \) holds significance.
• Part of the result stems from integrating a term aligned to the derivative of the denominator \( (2x + 2) \).

Logarithmic integration methods are vital, especially when a function or rational part aligns with the derivative structure inside a numerator simplifying integration.

Arctangent Integration

The arctangent integration arises especially in cases involving rational functions where the function forms a pattern that matches the derivative of \( \text{arctan}(x) \). It typically results when integrating expressions like \( \frac{1}{x^2 + a^2} \).

In the exercise, completing the square technique reveals that \( x^{2} + 2x + 2 \) translates to \( (x+1)^2 + 1 \). This matches the arctangent pattern:

• The integral \(2 \int \frac{1}{x^2 + 2x + 2} \, dx\) simplifies to \(2 \arctan(x+1)\).

Gaining fluency in recognizing and integrating these patterns is essential as the arctan function frequently appears in applications involving definite integrals and real-world problems.