Problem 48 - Calculus for Biology and Medicine

Question

In Problems 43-58, use substitution to evaluate each definite integral. $$ \int_{\ln 4}^{\ln 7} \frac{e^{x}}{\left(e^{x}+1\right)^{2}} d x $$

Step-by-Step Solution

Answer

The value of the integral is \( \frac{3}{40} \).

Step 1: Identify the Substitution

To simplify the integral, we will use the substitution method. Let \( u = e^x + 1 \). Then, differentiate \( u \) with respect to \( x \) to find \( du \), which gives \( du = e^x \, dx \).

Step 2: Rewrite the Integral with Substitution

Substitute \( u = e^x + 1 \) and \( du = e^x \, dx \) into the integral. The limits of integration change from \( x = \ln 4 \) to \( x = \ln 7 \):- When \( x = \ln 4 \), \( u = e^{\ln 4} + 1 = 4 + 1 = 5 \).- When \( x = \ln 7 \), \( u = e^{\ln 7} + 1 = 7 + 1 = 8 \).The integral now becomes \( \int_{5}^{8} \frac{1}{u^2} \, du \).

Step 3: Integrate with Respect to \( u \)

Integrate \( \int \frac{1}{u^2} \, du \). The antiderivative of \( \frac{1}{u^2} \) is \( -\frac{1}{u} \). So the integral becomes\[ \left[ -\frac{1}{u} \right]_{5}^{8} \].

Step 4: Evaluate the Definite Integral

Substitute the limits of integration into the antiderivative:\[ \left[ -\frac{1}{u} \right]_{5}^{8} = -\frac{1}{8} - \left( -\frac{1}{5} \right) = -\frac{1}{8} + \frac{1}{5} \].Simplify the expression to find the definite integral.

Step 5: Simplify the Result

Find a common denominator and simplify:\( -\frac{1}{8} + \frac{1}{5} = -\frac{5}{40} + \frac{8}{40} = \frac{3}{40} \).

Key Concepts

Substitution Method

The substitution method is a powerful technique used in calculus to evaluate certain integrals more easily. Essentially, it involves replacing a complicated expression within an integral with a single variable, referred to as the substitution variable. In this exercise, we identified the substitution by letting the expression inside the integral, that is accompanied by the most complexity, equal to a new variable. Here, we selected \( u = e^x + 1 \).

This substitution helps simplify the integral by transforming it into a form that is much easier to integrate. The next crucial step is to determine \( du \), which connects the original variable \( x \) to the new substitution variable \( u \). This involves differentiating \( u \) with respect to \( x \), which gives us \( du = e^x \cdot dx \).

• Choose the substitution variable wisely to simplify the integration process.
• Differentiate to find \( du \) so you can replace \( dx \) in the integral.
• Change the limits of the integral if it is definite, as we did by converting \( x = \ln 4 \) to \( u = 5 \) and \( x = \ln 7 \) to \( u = 8 \).

Change of Variables

Changing variables in an integral is much like re-routing a path to find an easier way to your destination. In the context of a definite integral, this change requires us to adjust not only the function being integrated but also the limits of integration. This adjustment is critical and often involves expressions that are algebraically simpler and hence, more easily manipulated.

In this problem, once we've completed our substitution and obtained \( u = e^x + 1 \), we need to determine the new integration limits. Integral limits \( x = \ln 4 \) and \( x = \ln 7 \) were recalculated under \( u \), transforming to \( u = 5 \) and \( u = 8 \).

After these adjustments, the integral \( \int_{\ln 4}^{\ln 7} \frac{e^x}{(e^x + 1)^2} dx \) becomes \( \int_{5}^{8} \frac{1}{u^2} \, du \). This step is crucial because without proper limits, your results won’t align with the physical or contextual meaning of the integral.

• Always change limits according to the new substitution variable.
• Ensure that both the integrand and integration limits are updated.
• Check calculations by simplifying the expression to confirm accuracy.

Antiderivative

The concept of the antiderivative is central to solving integrals. An antiderivative of a function is another function that, when differentiated, gives back the original function. Finding antiderivatives allows us to solve integrals, as they represent the reverse operation of differentiation.

For our integral \( \int \frac{1}{u^2} \, du \), the antiderivative is \( -\frac{1}{u} \). By evaluating this from \( u = 5 \) to \( u = 8 \), we obtain the result of the definite integral. Namely, \[ \left[ -\frac{1}{u} \right]_{5}^{8} = -\frac{1}{8} - \left(-\frac{1}{5} \right) = \frac{3}{40} \]
Understanding how to compute an antiderivative is essential, especially when encountering power functions, exponential functions, and trigonometric functions. Each type of function has its set of rules for finding the antiderivative.

• Know the rules for basic antiderivatives of different functions.
• Practice recognizing which antiderivative rule applies in a given situation.
• Calculate carefully while substituting and applying limits to avoid errors.