91Ó°ÊÓ

Integration involving logarithmic functions is known as logarithmic integration. When an integrand has a structure that resembles the derivative of a logarithm, it suggests a natural connection to logarithmic functions. For example, consider the integral \[ \int \frac{1}{x} \, dx \]This is directly integrable to \[ \ln|x| + C \]where \(C\) is the constant of integration. In the exercise provided, we deal with a similar scenario but with an exponential function in the denominator. By employing a suitable substitution to transform the denominator into a form that is directly integrable using logarithmic integration, we can find the indefinite integral.

U-substitution is a technique that simplifies integration by making a substitution that converts a difficult integral into an easier one. Essentially, it is akin to the reverse process of applying the chain rule in differentiation. The method involves identifying a part of the integrand that can be set as \( u \) — this is often a function within a function or a derivative that appears in the numerator. By substituting \( dx \) with \( du \), which is obtained by differentiating \( u \), the integral is translated into terms of \( u \), making it manageable. After integrating with respect to \( u \), we revert back to the original variable by substituting \( u \) back. In our exercise, setting \( u = 1 + e^x \) simplifies the integral into a form where the properties of logarithms can be easily applied.

The natural logarithm is a logarithm in base \( e \), where \( e \) is Euler's number, approximately equal to 2.71828. It is denoted by \( \ln(x) \) and is the inverse function of the exponential function \( e^x \). The natural logarithm has a unique property that the derivative of \( \ln|x| \) with respect to \( x \) is \( 1/x \), and this forms the basis for many integrations. In the given exercise, once \( u \)-substitution is applied, integrating \( 1/u \) directly yields the natural logarithm of \( u \), showcasing the connection between logarithms and integration.

Numerous integration techniques are employed to solve various integrals. The choice of technique greatly depends on the form of the integrand. Common methods include substitution, which includes \( u \)-substitution, integration by parts, trigonometric integration, partial fractions, and others. Each technique simplifies the integration process in different scenarios. For instance, substitution is effective when there's a function within another function, or when a derivative of a function is present in the integrand. The exercise we are looking at demonstrates a clever use of \( u \)-substitution to transform the integrand into a simpler form over which we have established integration strategies, such as logarithmic integration.