Problem 11 - Essential Calculus

Question

Find the integral. $$ \int \frac{e^{2 x}}{4+e^{4 x}} d x $$

Step-by-Step Solution

Answer

\(\frac{1}{2} \arctan(e^{2x}/2) + C\)

Step 1: Select substitution

Recall that the main idea behind substitution is to turn a complicated integral into a simpler one. Here the substitution \(u = e^{2x}\) can be considered. Then the derivative of \(u\), \(du\), would be \(2 e^{2x} dx\), or rearranged we have \(dx = du / (2e^{2x}) = du/(2u)\). Substituting this into the main problem simplifies the integral.

Step 2: Substitute and simplify

Now substitute \(u\) and \(dx\) into the integral: \(\int \frac{u}{4+u^2} \cdot \frac{1}{2u} du\). The \(u\) in the numerator and denomerator cancel out providing a nicer form: \(\frac{1}{2}\int \frac{1}{4+u^2} du\).

Step 3: Evaluate the Integral

The integral \(\frac{1}{2}\int \frac{1}{4+u^2} du\) is in a standard form whose antiderivative is known. In this case, it is the inverse tangent function, yielding: \(\frac{1}{2} \arctan(u/2) + C\).

Step 4: Back substitute

Now substitute the original variable back into the result. This means replacing \(u\) with \(e^{2x}\), giving the final result: \(\frac{1}{2} \arctan(e^{2x}/2) + C\).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions might sound intimidating, but they're actually quite intuitive when you break them down. Think of them as reverse operations of regular trigonometric functions like sine, cosine, and tangent. Their function is to give you the angle whose trigonometric function would provide a particular value. For example, the inverse function of tangent is arctan, noted as \( \arctan \). When you see an integral like \( \int \frac{1}{a^2 + u^2} \, du \), it takes the form of the inverse tangent, and its solution is \( \arctan\left(\frac{u}{a}\right) + C \).

• Key Point: In integrals, recognizing the pattern that leads to inverse trigonometric functions is crucial for simplifying and solving them efficiently.
• The integral \( \frac{1}{2}\int \frac{1}{4+u^2} \, du \) directly uses this principle to reach the solution \( \frac{1}{2} \arctan(u/2) + C \).

Integration Techniques

Integration techniques are methods we use to find antiderivatives or integrals, especially when they are not immediately obvious. Different integrals require different techniques, each tailored to particular forms. Let's look at some common techniques that are often used:

• Direct Integration: Used when the integral's form is straightforward and easily recognizable from the standard table of integrals.
• Integration by Parts: Involves decomposing the integral into a product of functions that can be integrated separately.
• Substitution: Involves replacing a part of the integral with a new variable (like \( u \)) to simplify the process, which brings us to our next section.

In our exercise, we used substitution, a powerful technique when dealing with complex expressions, especially those involving powers or composite functions. Remember, in integration, each technique you master becomes a tool in your math toolbox, making difficult problems easier.

Substitution Method

The substitution method is like solving a puzzle in mathematics. It helps simplify complex integrals by transforming them into more manageable forms. This technique is closely related to the chain rule in differentiation, where you replace one part of the expression with a new variable. Here's how it works in steps:

• Choose a Substitution: Identify a part of the integral that, when substituted, will simplify the expression. In our exercise, \( u = e^{2x} \) was the substitution that simplified the problem.
• Find the Differential: Compute \( du \) in terms of the original variable and differential. Here, \( dx = \frac{du}{2u} \).
• Substitute and Simplify: Replace the original variables with \( u \) and rewrite the integral. This might include cancelling terms, as we did when simplifying \( \frac{u}{2u} \).
• Integrate: Perform the simpler integral, which in our exercise became \( \frac{1}{2} \int \frac{1}{4+u^2} \ du \).
• Back-Substitute: Replace \( u \) with the original expression in terms of \( x \). This provides the answer in the original variable space.

The essence of substitution is to make integrals that look daunting much more approachable, allowing you to apply known solutions to basic integration scenarios.