Problem 30 - Calculus Single Variable

Question

Use the method of substitution to evaluate the definite integrals. $$ \int_{0}^{1} 24 \frac{\exp (x)}{(1+\exp (x))^{2}} d x $$

Step-by-Step Solution

Answer

\( \frac{12(e - 1)}{e + 1} \)

Step 1: Choose a Substitution

Notice that the expression \( 1 + \exp(x) \) appears to be a good candidate for substitution due to its presence in the denominator. Let's choose \( u = 1 + \exp(x) \).

Step 2: Differentiate the Substitution

To proceed, find \( \frac{du}{dx} \) which means we differentiate \( 1 + \exp(x) \) with respect to \( x \) to get \( \frac{du}{dx} = \exp(x) \). Therefore, \( du = \exp(x) \, dx \).

Step 3: Change the Limits of Integration

Convert the limits of integration from \( x \) to \( u \). When \( x = 0 \), \( u = 1 + \exp(0) = 2 \). When \( x = 1 \), \( u = 1 + \exp(1) = 1 + e \). So, the new limits for \( u \) are from 2 to \( 1 + e \).

Step 4: Express Integral in Terms of u

Substitute the expressions for \( u \) and \( du \) into the integral. The original integral becomes: \[\int_{2}^{1+e} 24 \cdot \frac{1}{u^2} \, du \]

Step 5: Integrate with Respect to u

Evaluate the integral: \[24 \int_{2}^{1+e} u^{-2} \, du = 24 \left[ -u^{-1} \right]_{2}^{1+e} \]This simplifies to \[-24 \left[ \frac{1}{1+e} - \frac{1}{2} \right]\]

Step 6: Simplify the Result

Now, compute the final value: \[-24 \left( \frac{1}{1+e} - \frac{1}{2} \right) = 24 \left( \frac{1}{2} - \frac{1}{1+e} \right)\]This can also be written as: \[24 \cdot \left( \frac{1 + e - 2}{2(1 + e)} \right) = 24 \cdot \frac{e - 1}{2(1 + e)} = \frac{24(e - 1)}{2(e + 1)} = \frac{12(e - 1)}{e + 1}\]

Step 7: Finalize the Answer

The integral evaluates to \( \frac{12(e - 1)}{e + 1} \) after simplification.

Key Concepts

Definite Integral

A definite integral is a fundamental concept in calculus that helps us compute the area under a curve from one point to another. In our example, we evaluated the integral \( \int_{0}^{1} 24 \frac{\exp (x)}{(1+\exp (x))^{2}} \, dx \). This process involves two main parts:

• The integrand, which is the function we want to integrate, here \( 24 \frac{\exp(x)}{(1+\exp(x))^2} \).
• The limits of integration, which in this case are from \( 0 \) to \( 1 \), representing the interval on the x-axis over which we are integrating.

The calculation of a definite integral provides us a specific numerical value, which contrasts with indefinite integrals that yield a family of functions plus a constant. For the integral above, the calculated area under the curve, after evaluating, is \( \frac{12(e - 1)}{e + 1} \). Remember, definite integrals have application in various fields such as physics, engineering, and probability, providing calculable results for continuous functions.

Exponential Functions

Exponential functions are one of the most important types of functions in mathematics. These functions involve the constant \( e \), approximately equal to 2.718, which is the base of the natural logarithms. An example of an exponential function is \( \exp(x) \), shorthand for \( e^x \). In the context of our integral, the expression involves \( \exp(x) \) in both the numerator and the denominator of the fraction.
Exponential functions are characterized by their unique properties such as:

• Having a constant rate of growth, which makes them ideal for modeling compounds like interest rates or population growth.
• Being always positive, thus they do not cross the x-axis.
• Having a specific derivative, \( \frac{d}{dx} e^x = e^x \), which means it is the only function whose derivative is identical to the function itself.

Understanding these properties helps when evaluating integrals that include exponentials, like in our example, making these concepts applicable in various scientific and mathematical contexts.

Integration by Substitution

Integration by substitution, also known as u-substitution, is an indispensable technique in calculus, used to simplify the integration process when dealing with complex functions. It involves substituting part of the integrand with a new variable to make the integration easier.
In our case, we chose the substitution \( u = 1 + \exp(x) \) to simplify the given integral. This choice was strategic because it turns a complex expression into a more manageable form. The fundamental steps include:

• Choosing a substitution \( u \), often based on the most complex part of the integrand or a component where its derivative also appears in the integrand.
• Taking the differential \( du = \exp(x) \, dx \).
• Converting the limits according to \( u \).
• Rewriting the integral in terms of \( u \), allowing us to solve a simpler integral.

By selecting the right substitution, complex integrals can be broken down into basic integral forms that are easier to evaluate. This technique is emphasized in calculus due to its versatility in handling a variety of problems.

Change of Variables in Integrals

The change of variables method is crucial for evaluating integrals effectively. This method simplifies the process of integration by converting the integrand into a form that is easier to handle.
In this scenario, we changed variables by assigning \( u = 1 + \exp(x) \). This requires the transformation of both the integrand and the limits of integration to the new variable \( u \). After the substitution, the original integral's limits changed from 0 to 1 in \( x \) to 2 to \( 1 + e \) in \( u \), making it easier to integrate.

• The change of variables is motivated by the presence of \( 1 + \exp(x) \) in both the integrand and its derivative.
• Transforming limits ensures the definite integral results remain consistent after the substitution.

This systematic approach is powerful as it often reduces the complexity of integration problems, providing a numerical solution more efficiently. The technique highlights the interconnected nature of calculus concepts and their capacity to solve complex real-world problems.