Problem 54 - Calculus for Biology and Medicine

Question

Use substitution to evaluate the definite integrals. $$ \int_{0}^{2} \frac{x}{x+2} d x $$

Step-by-Step Solution

Answer

The value of the definite integral is \( 2 - 2 \ln(2) \).

Step 1: Choose the Substitution

To perform substitution, we need to select a substitution that simplifies the integral. Let's choose \( u = x + 2 \). Then, the derivative of \( u \) with respect to \( x \) is \( \frac{du}{dx} = 1 \) or \( du = dx \).

Step 2: Update Integral Limits

Since \( u = x + 2 \), we need to change the limits of integration. When \( x = 0 \), \( u = 0 + 2 = 2 \). When \( x = 2 \), \( u = 2 + 2 = 4 \). So, the new limits for the integral are from \( u = 2 \) to \( u = 4 \).

Step 3: Substitute and Simplify the Integral

Substitute \( x = u - 2 \) and \( dx = du \) into the integral. The integral becomes: \[ \int_{2}^{4} \frac{u-2}{u} du \]. Simplify the expression to \( \int_{2}^{4} \left( 1 - \frac{2}{u} \right) du \).

Step 4: Evaluate the Simplified Integral

Now, evaluate \( \int_{2}^{4} \left( 1 - \frac{2}{u} \right) du \). This integral can be split into two separate integrals: \( \int_{2}^{4} 1 \ du - 2 \int_{2}^{4} \frac{1}{u} du \).

Step 5: Solve Each Part of the Integral

Evaluate the first integral: \( \int_{2}^{4} 1 \, du = [u]_{2}^{4} = 4 - 2 = 2 \). For the second integral, \( \int_{2}^{4} \frac{1}{u} \, du = [\ln |u|]_{2}^{4} = \ln 4 - \ln 2 = \ln(2) \).

Step 6: Combine the Results

Combine the results from Step 5: \( \int_{0}^{2} \frac{x}{x+2} \, dx = 2 - 2 (\ln(2)) \). The final answer is \( 2 - 2 \ln(2) \).

Key Concepts

Substitution Method

In calculus, the substitution method is an incredibly useful technique for evaluating integrals that may initially appear complex. This method involves selecting a new variable, often denoted as \( u \), which represents a part of the integral. The aim is to simplify the integral into a form that is easier to evaluate.When using substitution, one must:

• Choose a substitution that simplifies the integral.
• Derive the differential of the new variable \( du \) in terms of the original variable.
• Replace the integral bounds to fit the new variable \( u \).
• Rewrite the integral in terms of \( u \) and evaluate it.

In our example, we chose \( u = x + 2 \) to transform the integral into a simpler form. This substitution makes it possible to proceed with integrating using basic integration rules instead of facing a more difficult integral directly.

Calculus for Biology

Calculus has a variety of applications in the biological sciences. It's used for modeling and understanding dynamic systems such as population growth, rates of change in enzyme reactions, or the spread of diseases. With definite integrals, biologists can:

• Calculate areas under curves that represent biological phenomena like growth curves.
• Model biological systems and predict changes over time.
• Estimate rates of processes like nutrient absorption or medication decay in the body.

In biology, understanding the fundamental connections between change and accumulation helps scientists make predictions and explain biological phenomena with precision.

Integral Limits

Integral limits are the bounds within which the integration is performed. This refers to evaluating the function from one point to another, which provides the total accumulated value over that interval.In definite integrals:

• The lower limit represents where the accumulation begins.
• The upper limit represents where the accumulation ends.
• Integration changes these limits based on any substitutions made, ensuring that the new integral remains equivalent to the original.

Transforming these limits correctly is crucial. In our exercise, the limits were translated from \( x = 0 \) and \( x = 2 \) to \( u = 2 \) and \( u = 4 \) once the substitution \( u = x + 2 \) was applied, ensuring accurate results.

Logarithmic Integration

Logarithmic integration is a technique used when the integral involves logarithmic functions, often seen when integrating expressions like \( \frac{1}{x} \). This technique allows efficient handling of natural logs that can frequently appear in more complex integrals.To evaluate a logarithmic integral:

• Recognize the form \( \int \frac{1}{u} \, du = \ln |u| + C \).
• Apply this directly to evaluate the integral where such a form arises.
• Combine any constants and initial values appropriately to complete the calculation.

In the exercise, the integral \( \int_{2}^{4} \frac{1}{u} \, du \) simplifies to \( \ln |u| \). This logarithmic evaluation was a key part of finding the final solution \( 2 - 2 \ln(2) \). Mastering logarithmic integration can greatly simplify the process of tackling various integrals in calculus.