Question

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals. $$\int_{0}^{\ln 4} \frac{e^{x}}{3+2 e^{x}} d x$$

Short answer

Question: Evaluate the definite integral of the function $$\frac{e^x}{3+2e^x}$$ with respect to x over the interval [0, ln 4]. Answer: The definite integral is evaluated to be $$\frac{1}{2} [\ln(11) - \ln(5)]$$.

Step-by-step solution

Choose the substitution

Let's choose the substitution $$u = 3+2e^x$$. This choice will allow us to simplify the integral significantly.

Calculate the differential

Now, we need to find the differential (du) in terms of dx. Differentiating u with respect to x, we get: $$\frac{d u}{d x} = \frac{d(3+2e^x)}{dx} = 2e^x$$ Now, let's solve for dx: $$d x = \frac{d u}{2 e^{x}}$$

Substitute u and du into the integral

Now, we'll substitute u and du into the original integral and change the limits of integration accordingly. When $$x = 0$$, $$u = 3+2e^0 = 3+2 = 5$$ When $$x = \ln 4$$, $$u = 3+2e^{\ln 4} = 3+2(4) = 11$$ So, the updated integral becomes: $$\int_{5}^{11} \frac{1}{u} \cdot \frac{d u}{2}$$

Evaluate the new integral

Now, we can evaluate the new integral: $$\int_{5}^{11} \frac{1}{u} \cdot \frac{d u}{2} = \frac{1}{2} \int_{5}^{11} \frac{1}{u}\, du$$ Using the property that $$\int \frac{1}{u}\, du = \ln |u| + C$$, we can evaluate the definite integral as: $$\frac{1}{2} \left[\ln |u| \right]_{5}^{11} = \frac{1}{2} [\ln(11) - \ln(5)]$$

Substitute back the original variable

Now, let's substitute u back in terms of x: $$\frac{1}{2} [\ln(11) - \ln(5)] = \frac{1}{2} [\ln(3+2e^{\ln 4}) - \ln(3+2e^0)] = \frac{1}{2} [\ln(11) - \ln(5)]$$ The definite integral is evaluated to be $$\frac{1}{2} [\ln(11) - \ln(5)]$$.

Key concepts

Change of Variables

When dealing with definite integrals, the change of variables technique, also known as u-substitution, allows us to simplify complex integrals by transforming them into easier ones. This method is akin to the chain rule in differentiation and involves substituting a part of the integral with a single variable, say, \( u \).

The goal of this technique is to rewrite the integral in terms of \(u\) instead of \(x\). The steps include:

• Choosing a substitution \( u \) that simplifies the integral.
• Calculating the differential \( du \) in terms of the original variable \( dx \).
• Substituting both \( u \) and \( du \) into the integral and adjusting the integration limits.

These steps help in transforming a complicated integral into a simpler one, making it easier to evaluate. Once the new integral is solved, the last step is to return to the original variable \( x \) and ensure the evaluated result matches the context of the original problem.

In our problem, we used \( u = 3+2e^x \), simplifying the original integral into an easily solvable form. This choice efficiently reduced the complexity of dealing with an exponential function and transformed it into a logarithmic integral.

Integration Techniques

Various integration techniques can help solve different types of integrals. One such valuable technique is using logarithmic properties in integration when the integrand includes rational expressions, such as \( \frac{1}{u} \).

For integrals of the form \( \int \frac{1}{u} \, du \), the solution is derived as \( \ln |u| + C \), where \( C \) is the integration constant. This property allows us to express the result of the integration in terms of natural logarithms, simplifying the process.

In the given example, after substituting and simplifying, the integral \( \int \frac{1}{u} \, du \) is straightforward to evaluate using this fundamental property. Always remember, when evaluating definite integrals, the integral limits must also be converted to match the variable of substitution.

Thus, with our limits changed along with \( u \), the integral was solved simply as \( \frac{1}{2} [\ln(11) - \ln(5)] \), demonstrating a clear use of logarithms in effectively solving integrals that might seem daunting at first.

Transcendental Functions

In the realm of calculus, transcendental functions do not arise from simple algebraic operations. They include functions such as exponential, logarithmic, and trigonometric functions.

In this exercise, the function \( e^x \) is transcendental. Evaluating integrals with transcendental functions often requires more than basic integration rules. These functions tend to necessitate advanced techniques like substitution to simplify and resolve such integrals.

For example, the substitution \( u = 3 + 2e^x \) helped deal with the \( e^x \) component in our integral. After substitution, the seemingly complex transcendental function was reduced to a simpler logarithmic function, \( \ln |u| \). This transformation brings out the power of understanding transcendental functions and applying appropriate techniques to handle their integrals successfully.

So, while transcendental functions may appear intricate, understanding their nature and behavior can vastly simplify problems through strategic approaches like the one used in this solution.