Problem 13 - Calculus for Biology and Medicine

Question

In Problems 1-16, evaluate each 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

The indefinite integral is \( \ln |x^2 + 4x| + C \).

Step 1: Identify the Substitution

We start by noting the substitution provided in the exercise: let \( u = x^2 + 4x \). Our goal is to rewrite the integral in terms of \( u \).

Step 2: Differentiate the Substitution

We need to find \( du \) in terms of \( dx \). Differentiate \( u = x^2 + 4x \) with respect to \( x \):\[ du = (2x + 4) dx \].

Step 3: Solve for dx

From the expression \( du = (2x + 4) dx \), solve for \( dx \):\[ dx = \frac{du}{2x + 4} \].

Step 4: Rewrite the Integrand

Substitute \( dx = \frac{du}{2x + 4} \) into the integral:\[ \int \frac{x+2}{u} \cdot \frac{du}{2x + 4} \]. Note that \( x + 2 = \frac{(x^2 + 4x)'}{2} \), thus \( x + 2 = \frac{2x + 4}{2} \) simplifies to \( 1 \).

Step 5: Simplify the Integral

Recognizing that \( x+2 = \frac{2x+4}{2} \), the integral simplifies to:\[ \int \frac{du}{u} \]. This is a standard integral form.

Step 6: Integrate

Evaluate the integral:\[ \int \frac{1}{u} \, du = \ln |u| + C \] where \( C \) is the constant of integration.

Step 7: Back-Substitute for x

Substitute back \( u = x^2 + 4x \) into the expression:\[ \ln |x^2 + 4x| + C \].

Key Concepts

Integration by Substitution

Integration by substitution is a technique used to simplify the process of evaluating integrals. It involves transforming a complex integral into a simpler form through a substitution. This method is particularly useful when you encounter integrals that contain composite functions.
The main steps in the substitution method are:

• Choose a substitution: Identify part of the integrand that can be replaced with a new variable, typically referred to as `u`.
• Differentiate: Calculate the differential `du` in terms of `dx` to facilitate the conversion of the entire integral into terms of `u`.
• Substitute: Replace the identified part of the integrand and `dx` with `u` and `du`, respectively, to obtain a new integral that's often easier to solve.

By using substitution in our given exercise, we convert the more complicated form into  a much cleaner and simpler integral, allowing for straightforward evaluation.

Differential Calculus

Differential calculus is a fundamental concept in mathematics focusing on the concept of a derivative, which quantifies how a function changes as its input changes.
When employing substitution in integration, you use differential calculus to find the differentials in terms of each variable.

For example, in our exercise, we have the substitution:

• Define the new variable: Let \( u = x^2 + 4x \).
• Differentiate: Compute the derivative of \( u \) with respect to \( x \), which gives you \( du = (2x + 4) dx \).
• Solve for \( dx \): Rearrange to find \( dx \) in terms of \( du \), which is \( dx = \frac{du}{2x+4} \).

Expressing \( du \) and \( dx \) is crucial because it allows us to reframe the integral in a manner that is algebraically conducive for integration. With these derivatives in hand, you're equipped to transform the integral into one involving \( u \), making it much simpler to solve.

Integration Techniques

Integration involves a variety of techniques, each designed to make the evaluation of integrals more manageable. Understanding when to apply each technique is key to successful problem-solving.

Some common integration techniques include:

• Basic integration: Finding the antiderivative directly from the standard forms.
• Integration by parts: Dealing with products of functions by splitting them into more manageable pieces.
• Integration by substitution: Simplifying the integral by changing its variable, as seen in our exercise.
• Trigonometric substitution: Used for integrals involving roots of expressions, often involving trigonometric identities.

Each technique has its own purpose and applies to specific forms of functions within the integral. In practice, solving integrals often involves trying different methods until a solution emerges. For our example, after substitution simplifies the integral, it fits a standard form, leading to the result \( \int \frac{1}{u} \, du = \ln |u| + C \). The power of these techniques lies in transforming complex integrands into forms that can be readily integrated.