Problem 38 - College Algebra and Calculus: An Applied Approach

Question

Use a symbolic integration utility to find the indefinite integral. $$ \int \frac{-e^{3 x}}{2-e^{3 x}} d x $$

Step-by-Step Solution

Answer

The indefinite integral of \(\int \frac{-e^{3 x}}{2-e^{3 x}} d x\) is \(\frac{1}{3} \ln|2 - e^{3x}|\).

Step 1: Identify a suitable substitution

A common strategy when dealing with integrals involving exponential functions is to use substitution. In this case, the denominator \(2 - e^{3x}\) can be a suitable choice for substitution. Therefore, let \(u = 2 - e^{3x}\). This will simplify the integral considerably.

Step 2: Find the derivative of the substituting function

Next we need to find the derivative of \(u\), which we denote as \(du\). The derivative of \(2 - e^{3x}\) is \(-3 e^{3x}\). Therefore, \(du = -3e^{3x} dx\). Now, if we solve this for \(dx\), we get \(dx = -\frac{du}{3e^{3x}}\).

Step 3: Substitute into the integral

With these substitutions, we can now rewrite the original integral. It becomes \(\int \frac{-e^{3x}}{u} \times -\frac{du}{3e^{3x}} \), simplifying to \(\int \frac{du}{3u}\).

Step 4: Simplify and integrate

The new integral from Step 3 is easier to solve, it's \(\frac{1}{3} \int \frac{du}{u}\). The result of this integration according to integral rules is \(\frac{1}{3} \ln|u|\).

Step 5: Substitute back the original variable

Substitute \(u = 2 - e^{3x}\) back into the integral, the final solution is \(\frac{1}{3} \ln|2 - e^{3x}|\).

Key Concepts

Symbolic Integration

Symbolic integration refers to the mathematical process of finding antiderivatives. It's a core component of calculus, which, unlike numerical integration that provides an approximation, yields exact expressions for the integral of a function. This is particularly useful when dealing with indefinite integrals, which represent the family of all antiderivatives of a given function.

For instance, when you encounter an integral like \(\int \frac{-e^{3 x}}{2-e^{3 x}} dx\), a symbolic integration utility would be instrumental in finding a precise, closed-form expression for its antiderivative. High-level understanding of symbolic integration is beneficial because it opens up paths for further mathematical manipulations and applications, including series expansions and evaluation of definite integrals.

Exponential Functions

Exponential functions, represented as \(e^x\) where \(e\) is the base of the natural logarithm, have unique properties that are central to many areas of mathematics, including calculus. The function \(e^x\) is its own derivative, which makes it extraordinarily manageable when it comes to differentiation and integration.

With indefinite integrals involving exponential functions, one often needs an effective approach to simplify the integral before finding a solution. Keep in mind that every exponential function, like \(e^{3x}\), grows or decays exponentially, and this property influences how they behave under integration, making techniques like substitution often necessary to solve them.

Integration by Substitution

Integration by substitution, sometimes referred to as the reverse chain rule, is a technique to simplify integrals by changing variables. It's akin to performing a 'u-turn' to make the integral more manageable. Consider the integral \(\int \frac{-e^{3 x}}{2-e^{3 x}} dx\); a direct approach is not straightforward.

However, by substituting a part of the integrand with a new variable \(u\), the complexity can be significantly reduced. This process involves identifying a function within the integrand whose derivative is also present or can be made present by simple manipulation. With a keen eye, one can often spot a potential substitution which simplifies and streamlines the integration process.

U-Substitution

U-substitution is a specific application of integration by substitution. This strategy involves choosing a part of the integrand, assigning it to the variable \(u\), and then expressing the differential \(dx\) in terms of \(du\).

The goal of u-substitution is to transform the integral into a simpler form, which often reverts the problem to a basic integral that is easily solvable. As in the example \(\int \frac{-e^{3 x}}{2-e^{3 x}} dx\), you would let \(u = 2 - e^{3x}\) and find \(du = -3e^{3x}dx\), which upon solving for \(dx\) yields \(dx = -\frac{du}{3e^{3x}}\). Substituting these into the original integral simplifies it, leading to an easily integrable expression in terms of \(u\). This elegance of u-substitution makes it a staple technique in solving integrals.