Problem 69 - Calculus Single Variable

Question

Calculate the integrals. $$ \int \frac{\exp (x)}{1+\exp (2 x)} d x $$

Step-by-Step Solution

Answer

The integral is \( \arctan(\exp(x)) + C \).

Step 1: Let the Substitution Take Over

First, let's use a substitution to simplify the integral. Set \( u = \exp(x) \). Then the derivative of \( u \) with respect to \( x \) is \( du = \exp(x) \, dx \). This transforms our integral into \( \int \frac{1}{1 + u^2} \, du \).

Step 2: Recognize the New Form

The integral \( \int \frac{1}{1 + u^2} \, du \) is a standard integral that corresponds to the arctangent function. This is a well-known result that computes to \( \arctan(u) + C \), where \( C \) is the constant of integration.

Step 3: Reverse the Substitution

Replace \( u \) back with \( \exp(x) \) to complete the solution in terms of the original variable. Thus, we substitute \( u = \exp(x) \) back into the solution, resulting in \( \arctan(\exp(x)) + C \).

Key Concepts

Substitution Method

In calculus, the substitution method is a powerful technique used to simplify complex integrals by introducing a new variable. This method essentially involves substituting a part of the integral with a single variable to reduce its complexity. In our example, we substitute \( u = \exp(x) \) for the given integral \( \int \frac{\exp(x)}{1+\exp(2x)} \, dx \).

• First, notice the appearance of \( \exp(x) \) in both the numerator and denominator of the integrand. Setting \( u = \exp(x) \) simplifies the expression significantly.
• To change the variable of integration from \( x \) to \( u \), find the derivative: \( du = \exp(x) \, dx \), which directly replaces \( \exp(x) \, dx \) in our integral.
• Upon substitution, the integral becomes \( \int \frac{1}{1 + u^2} \, du \), which is much more manageable.

The beauty of the substitution method lies in its ability to transform an intricate problem into a recognizably simpler form, making it easier to integrate.

Arctangent Function

Upon transforming the integral using substitution, we encounter the integral \( \int \frac{1}{1+u^2} \, du \). This is a classical form that directly correlates with the arctangent function.

• The integral \( \int \frac{1}{1+u^2} \, du \) equates to \( \arctan(u) + C \), where \( C \) is the integration constant.
• The arctangent function, \( \arctan(x) \), is the inverse of the tangent function, and it maps real numbers to angles, specifically in radians.
• In practical terms, when you see \( \frac{1}{1+u^2} \), it should trigger a recognition of the arctangent function as its antiderivative.

Understanding the connections between integral expressions and trigonometric functions can significantly streamline solving integrals and make these problems more intuitive.

Exponential Functions

Exponential functions, denoted as \( \exp(x) \) or \( e^x \), are fundamental in calculus and appear frequently in integration. They have unique properties that make them particularly interesting in mathematics.

• The exponential function \( \exp(x) \) grows rapidly and is defined as its own derivative, meaning \( \frac{d}{dx}[\exp(x)] = \exp(x) \).
• In the context of our integral \( \int \frac{\exp(x)}{1+\exp(2x)} \, dx \), \( \exp(x) \) appears in both the numerator and denominator, making it an ideal candidate for substitution.
• The substitution simply replaces \( \exp(x) \) with \( u \), taking advantage of its derivative being \( \exp(x) \, dx \), simplifying integration considerably.

Grasping the characteristics of exponential functions can empower you to handle complex integrals more efficiently, and they are instrumental in various fields such as compound interest, population dynamics, and natural phenomena modeling.