Problem 65 - Calculus for Biology and Medicine

Question

Evaluate each integral. $$ \int \frac{2 x^{2}+x+5}{x\left(x^{2}+2 x+5\right)} d x $$

Step-by-Step Solution

Answer

\[ \ln|x| + \frac{1}{2}\ln|x^2 + 2x + 5| - \frac{1}{\sqrt{21}} \tan^{-1}\left(\frac{x+1}{\sqrt{21}}\right) + C \]

Step 1: Simplify the Integrand

The first step in evaluating this integral is to simplify the given integrand. Notice that the expression \ \( \frac{2x^2 + x + 5}{x (x^2 + 2x + 5)} \) can be addressed by splitting it into partial fractions. We need to express it as \ \( \frac{A}{x} + \frac{Bx + C}{x^2 + 2x + 5} \).

Step 2: Set Up Partial Fraction Decomposition

We want to find constants \ A, B, \ and \ C \ such that: \ \( \frac{2x^2 + x + 5}{x(x^2 + 2x + 5)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 2x + 5} \). Equating the numerators, we get:\ \( 2x^2 + x + 5 = A(x^2 + 2x + 5) + (Bx + C)x \).

Step 3: Solve for Constants

Expand and equate the coefficients from both sides of the equation:\ \( = (A + B)x^2 + (2A + C)x + 5A \).Comparatively, this matches \ 2x^2 + x + 5 \, yielding the system:1. \ A + B = 2 \2. \ 2A + C = 1 \3. \ 5A = 5 \ Solving these gives \ A = 1, B = 1, C = -1 \.

Step 4: Rewrite the Integrand

Substitute back the values of A, B, and C into the partial fractions:\[ \int \left( \frac{1}{x} + \frac{x - 1}{x^2 + 2x + 5} \right) dx \].

Step 5: Integrate Each Term

Now, integrate term by term:1. \( \int \frac{1}{x} dx = \ln|x| + C_1 \).2. For \( \int \frac{x}{x^2 + 2x + 5} dx \), use substitution, let \( u = x^2 + 2x + 5 \), then integrate to find \ \( \frac{1}{2} \ln|x^2 + 2x + 5| + C_2 \).3. \( \int \frac{-1}{x^2 + 2x + 5} dx \) results in \ -\frac{1}{\sqrt{21}} \tan^{-1}\left(\frac{x+1}{\sqrt{21}}\right) + C_3 \.

Step 6: Combine Results

Combine the results from all integrals:\[ \ln|x| + \frac{1}{2}\ln|x^2 + 2x + 5| - \frac{1}{\sqrt{21}} \tan^{-1}\left(\frac{x+1}{\sqrt{21}}\right) + C \].

Key Concepts

Partial Fraction Decomposition

When tackling an integral that involves a rational function, partial fraction decomposition is a handy technique. It allows us to break down complex fractions into simpler, more manageable pieces, making integration much easier. This method is particularly useful when the integrand is a quotient of polynomials, as seen in our exercise.

In the given problem, the integrand is \( \frac{2x^2 + x + 5}{x(x^2 + 2x + 5)} \). We can split this into simpler parts by expressing it as \( \frac{A}{x} + \frac{Bx + C}{x^2 + 2x + 5} \).

• Here, \( A \), \( B \), and \( C \) are constants we need to determine.
• By equating coefficients after multiplying through by the denominator, we find these constants.

This technique reduces the complexity of the original problem, allowing each simpler piece to be integrated individually.

Integration Techniques

In integral calculus, using the right integration technique is crucial for solving integrals effectively. For the problem \( \int \frac{2 x^{2}+x+5}{x(x^{2}+2 x+5)} dx \), we simplify the integrand first using partial fraction decomposition.

The integration process here involves three main techniques:

• Direct integration of functions like \( \int \frac{1}{x} dx = \ln|x| \).
• Substitution method for integrals where the substitution \( u = x^2 + 2x + 5 \) helps simplify the expression.
• Recognizing forms that result in inverse trigonometric integrals, applicable for the term \( \int \frac{-1}{x^2 + 2x + 5} dx \).

Choosing the right technique is a critical step in efficiently handling challenging integrals.

Logarithmic Integration

Logarithmic integration is a technique used when dealing with integrals similar to \( \int \frac{1}{x} dx \), which results in a natural logarithm.

In our problem, after partial fraction decomposition, one term directly integrates to \( \ln|x| \). Another, more complex term, \( \int \frac{x}{x^2 + 2x + 5} dx \), can also fit this pattern when coupled with substitution.

• By letting \( u = x^2 + 2x + 5 \), the integral transforms into a simpler form, working towards \( \frac{1}{2}\ln|x^2 + 2x + 5| \).
• Logarithmic forms are common in integrals involving rational functions.

Mastering this technique enhances your ability to handle integrals featuring division by simple or complex expressions.

Substitution Method

The substitution method is a fundamental tool in calculus, particularly useful when dealing with integrals that are not easily tackled with basic formulas. It essentially involves changing the variable of integration to simplify the integral.

In our exercise, substitution plays a key role in integrating the term \( \int \frac{x}{x^2 + 2x + 5} dx \). By setting \( u = x^2 + 2x + 5 \), we reconfigure the integral:

• The derivative \( du = (2x + 2) dx \) suggests \( dx = \frac{du}{2x+2} \).
• This simplifies the integral into a form where standard techniques like logarithmic integration can be applied.

This method is a powerful way to transform and simplify problems, making complex integrals much easier to manage.