Question

Use the Log Rule to find the indefinite integral. $$ \int \frac{e^{-x}}{1-e^{-x}} d x $$

Short answer

The integral is equal to \( - \ln |1 - e^{-x}| + C \)

Step-by-step solution

Identify Substitution

For clear understanding, let's substitute \( e^{-x} = t \). Thus, \( -e^{-x} dx = dt \) or \( dx = -dt / t \). The integral then becomes \( - \int \frac{dt}{1 - t} \).

Apply Logarithmic Rule

We recognize the integral form as the logarithmic rule. The integral becomes \( - \ln |1 - t| + C \).

Substitute Back

Substitute back \( t = e^{-x} \) and thus the indefinite integral becomes \( - \ln |1 - e^{-x}| + C \)

Key concepts

Logarithmic Integration

Logarithmic integration is a method used when an integral involves a function divided by its derivative. In such cases, the logarithmic rule can be directly applied.

Integrals of the form \( \int \frac{f'(x)}{f(x)} dx \) are solved using the natural logarithm, due to the derivative of the natural logarithm being \( \frac{1}{x} \). The result is \( \ln |f(x)| + C \), where C is the integration constant. This approach is particularly useful when a function's logarithm can be easily obtained. If faced with \( \int \frac{e^{-x}}{1-e^{-x}} dx \), logically recognize the numerator as the derivative of the denominator's inside function and apply the logarithmic rule, making the solution \( -\ln |1-e^{-x}| + C \).

To master logarithmic integration, familiarize yourself with the properties of logarithms and practice by integrating various functions using this method.

Integration by Substitution

Integration by Substitution, often referred to as u-substitution, is a technique similar to the chain rule for differentiation, but applied in reverse for integration. It is primarily used when an integral contains a function and its derivative.

Consider an integral of the form \( \int g(f(x))f'(x) dx \), where u = f(x) and du = f'(x) dx. Substituting these values simplifies the integral to \( \int g(u) du \). Once integrated, substitute back the original f(x) to obtain the solution in terms of x.

For example, in the exercise \( \int \frac{e^{-x}}{1-e^{-x}} dx \), by letting t = e^{-x} and replacing accordingly, the integral simplifies, allowing easier integration. This shows why substitution can be an invaluable tool in solving complex integrals and emphasizes the importance of selecting the right substitution to simplify an integral as much as possible.

Exponential Functions

Exponential functions are a class of mathematical functions of the form \( f(x) = a^x \), where a is a positive constant called the base. The most commonly encountered base is e, the Euler's number, which is approximately 2.71828.

Exponential functions are unique as their rate of growth is proportional to their value. \( e^x \), in particular, is fundamental in calculus due to its interesting property of being the rate of its own change, meaning \( \frac{d}{dx}e^x = e^x \) and its integral \( \int e^x dx = e^x + C \). This makes e^x a perfect tool for solving a vast array of problems in mathematics and its applications, such as in the given exercise.

Studying the behavior and properties of exponential functions is essential for understanding growth and decay processes in natural and social sciences. Being comfortable with these concepts improves computational skills and problem-solving strategies in calculus.