Problem 13 - Calculus for Biology and Medicine

Question

Evaluate the indefinite integral by making the given substitution. $$ \int \frac{x+2}{x^{2}+4 x} d x, \text { with } u=x^{2}+4 x $$

Step-by-Step Solution

Answer

\( \int \frac{x+2}{x^{2}+4x} \, dx = \frac{1}{2} \ln|x^2 + 4x| + C \)

Step 1: Analyze the Integrand

The problem asks us to evaluate the integral \( \int \frac{x+2}{x^{2}+4x} \, dx \). We notice that the numerator, \( x+2 \), is related to the derivative of the denominator \( x^2 + 4x \).

Step 2: Perform Substitution

We make the substitution \( u = x^2 + 4x \). The derivative is \( \frac{du}{dx} = 2x + 4 \), hence \( du = (2x + 4) \, dx \) or \( du = 2(x + 2) \, dx \). Thus, \( dx = \frac{du}{2(x+2)} \).

Step 3: Substitute in the Integral

Substitute \( u \) and \( dx \) into the integral: \[\int \frac{x+2}{u} \cdot \frac{du}{2(x+2)} = \int \frac{du}{2u}\]

Step 4: Simplify the Integral

\( \int \frac{du}{2u} = \frac{1}{2} \int \frac{1}{u} \, du \). This is a standard logarithmic integral.

Step 5: Integrate

The integral \( \int \frac{1}{u} \, du \) results in \( \ln|u| + C \), where \( C \) is the constant of integration. Thus, \( \frac{1}{2} \ln|u| + C \) is the integrated form.

Step 6: Back-substitute for x

Replace \( u \) with \( x^2 + 4x \): \[\frac{1}{2} \ln|x^2 + 4x| + C\]

Key Concepts

Substitution Method

The substitution method is a technique used to simplify the process of finding an indefinite integral. This method involves replacing a complex expression within the integral with a single variable, making the integration process more straightforward. To use the substitution method effectively, follow these basic steps:

• Identify a part of the integrand that can be replaced by a simpler expression, typically one that its derivative is present elsewhere in the integrand.
• Substitute the chosen expression with a new variable, generally denoted by \( u \), and find \( du \) in terms of \( dx \).
• Rewrite the integral in terms of \( u \), which often simplifies the integral significantly.
• After integrating, remember to substitute back the original variable to express the result in terms of the initial variable.

This method transforms complicated integrals into more manageable forms, leveraging the chain rule in reverse.

Logarithmic Integral

The logarithmic integral arises frequently in calculus, especially when dealing with integrals involving fractions. It occurs most commonly with the format \( \int \frac{1}{u} \, du \), which computes to \( \ln|u| + C \), where \( C \) is the constant of integration.

This is a standard result due to the derivative of the natural logarithm \( \ln|x| \) being \( \frac{1}{x} \). When solving integrals, recognizing when the integrand is a derivative of the natural log function helps simplify integration to a result involving \( \ln \).

Don't forget the absolute value in the logarithmic integral, as the natural logarithm's domain only includes positive numbers.

Integration by Substitution

In our example, integration by substitution helped simplify the integral with the given substitution \( u = x^2 + 4x \). This technique is not only a powerful tool in recalibrating the integral into a simpler form but also useful when components of an integrand resemble derivatives of other parts.
By noting that \( x + 2 \) within the integrand was closely connected to the derivative of \( x^2 + 4x \), substitution allowed us to transform the original expression into a form ready for logarithmic integration:

• We rewrote \( dx \) in terms of \( du \).
• This adjustment turned the problem into a straightforward \( \int \frac{1}{u} \, du \), which integrated to a log function.
• Finally, we reverted back to the initial variable to complete the solution.

Integration by substitution is akin to reversing the chain rule and is invaluable for untangling complex integrals.