Problem 36 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

The indefinite integral of \(\int \frac{3 e^{x}}{2+e^{x}} dx\) is \(3 \ln|2+e^{x}| + C\)

Step 1: Simplifying the Integrant

First, let's simplify the integrand of the function. We can make a substitution: Let \( u = 2 + e^{x} \). Then the derivative of \( u \) with respect to \( x \) is \( du/dx = e^{x} \) or \( du = e^{x}dx \). Now the integral becomes: \( \int 3 \frac{du}{u} \)

Step 2: Integration

Now we can integrate the new function using the rule: \( \int \frac{du}{u} = \ln|u| \). The integral becomes: \( 3 \int \frac{du}{u} = 3\ln|u| \)

Step 3: Substituting back

Now we put \( u \) back into our integral, so our indefinite integral becomes: \( \int \frac{3 e^{x}}{2+e^{x}} dx = 3 \ln|2+e^{x}| + C \), where \( C \) is the constant of integration.

Key Concepts

Symbolic Integration

Symbolic integration is a fundamental concept in calculus, referring to the process of finding an antiderivative or indefinite integral of a function symbolically, rather than numerically. The goal is to express the antiderivative in a closed-form expression, which involves algebraic, trigonometric, exponential, or logarithmic functions.

Substitution Method

The substitution method, also known as 'u-substitution', is a powerful technique for simplifying integrals. By choosing a substitution for a part of the original integrand, it transforms a complex integral into a simpler one.

For the exercise given, we use the substitution method by setting a new variable, typically noted as 'u', equal to an expression found within our original integral. In this case, the substitution was made such that
$$ u = 2 + e^{x} $$.
With the derivative
$$ du = e^{x} dx $$.
This clever trick rewrites the integral in terms of 'u', which can then be integrated more easily.

Natural Logarithm

The natural logarithm, denoted as 'ln', is the logarithm to the base 'e', where 'e' is an irrational and transcendental constant approximately equal to 2.71828. In integration, the natural logarithm appears when the integral of
$$ \frac{1}{x} $$
is taken. The result is
$$ \ln |x| + C $$.

In the given exercise, after the substitution method is applied, integrating
$$ \frac{du}{u} $$
leads to
$$ \ln |u| $$, which is an application of this logarithmic integration rule.

Integration Rules

Integration encompasses a set of rules that guide the integration process, such as linearity, which allows us to integrate term-by-term, and the power rule. When faced with a fraction where the numerator is the derivative of the denominator, as in our exercise, we apply another rule that leads directly to the natural logarithm of the denominator. Understanding these rules and recognizing when to apply them is crucial for correctly solving integrals.

Constant of Integration

The constant of integration, often denoted by 'C', is a critical aspect of indefinite integrals. Whenever we integrate a function, there is an infinite number of possible antiderivatives, each differing by a constant. Thus, when we symbolically integrate a function and find its antiderivative, we add 'C' to represent this family of solutions.

In our exercise, after finding
$$ 3\ln|u| $$,
we must not forget to add 'C' to our final answer to indicate the most general form of the antiderivative.