Problem 14 - Calculus for Biology and Medicine

Question

Evaluate the indefinite integral by making the given substitution. $$ \int \frac{2 x}{3-x^{2}} d x, \text { with } u=3-x^{2} $$

Step-by-Step Solution

Answer

\(-\ln|3 - x^2| + C\)

Step 1: Identify the Substitution

We're given the substitution \( u = 3 - x^2 \). From this, calculate the derivative \( \frac{du}{dx} \). This is crucial to change the variables in the integral.

Step 2: Compute the Derivative

Take the derivative of \( u = 3 - x^2 \) with respect to \( x \):\[ \frac{du}{dx} = -2x. \]

Step 3: Rearrange for dx

From the derivative, you get: \[ du = -2x \, dx \] or rearranging gives \[ dx = \frac{du}{-2x}. \]

Step 4: Substitute in the Integral

Substitute \( u = 3 - x^2 \) and \( dx = \frac{du}{-2x} \) into the integral. The integral becomes: \[ \int \frac{2x}{u} \cdot \frac{du}{-2x}. \] The \( 2x \) terms cancel out.

Step 5: Simplify the Integral

The integral simplifies to: \[ -\int \frac{1}{u} \, du. \]

Step 6: Integrate with Respect to u

The integral of \( \frac{1}{u} \) with respect to \( u \) is the natural logarithm: \[ -\ln|u| + C. \] Where \( C \) is the constant of integration.

Step 7: Substitute Back the Original Variable

Since \( u = 3 - x^2 \), substitute back to obtain: \[ -\ln|3 - x^2| + C. \]

Key Concepts

Substitution Method

The substitution method is an essential tool in solving indefinite integrals, especially when dealing with complex expressions. When using this technique, the goal is to "substitute" part of the integral with a new variable, ideally simplifying the integral into a more recognizable form. Here's how it works:

• Identify a part of the integral that can be replaced with a new variable, say, **u**.
• Compute the derivative of **u** with respect to **x** to find **du/dx**.
• Rearrange this derivative into the form **du = ... dx**, allowing you to replace **dx** in the integral.
• Replace the identified part and **dx** in the original integral with **u** and **du**, respectively.
• Once substituted, the integral should be simpler and solvable with basic techniques.

In our problem, the substitution **u = 3 - x^2** was used, transforming the integral into an easier form involving **u**. This method frequently aids in simplifying expressions where direct integration seems challenging.

Integration Techniques

Integration techniques are diverse tools that help tackle the wide variety of forms indefinite integrals can take. Each technique is suitable for particular types of integrals. The most common techniques include:

• **Substitution Method:** As discussed, this technique simplifies an integral by changing variables.
• **Integration by Parts:** Useful when dealing with products of functions. It's based on the product rule for derivatives.
• **Partial Fractions:** Decomposes complex rational expressions into simpler ones. Particularly helpful with quadratic or cubic denominators.
• **Trigonometric Identities:** Utilizes trigonometric formulas to simplify integrals involving sin, cos, and other trigonometric functions.

In our example, the substitution method was the ideal choice due to its ability to simplify the expression from one involving **x** to a much simpler one in **u** that could then be integrated directly.

Natural Logarithm

The natural logarithm, denoted as **ln**, is the logarithm to the base **e**, where **e** is approximately 2.71828. It's a cornerstone of calculus, appearing frequently in integration, especially with expressions of the form **1/u**.

• When integrating **1/u**, the result is the natural logarithm function, **ln|u| + C**, where **C** is the constant of integration.
• This is because the derivative of **ln|u|** with respect to **u** is exactly **1/u**.
• Understanding the natural logarithm is important, as it helps to solve integrals that do not fit the standard basic patterns.

In our exercise, after substitution, the integral simplifies to **-ln|u| + C**. Substituting **u** back, gives **-ln|3 - x^2| + C**. Thus, recognizing when an integral results in a natural logarithm is a significant piece of mastering integration.