Question

Find the antiderivative. \(\int \frac{e^{2 x}}{1+e^{4 x}} d x\)

Short answer

The antiderivative is \( \frac{1}{2} \arctan(e^{2x}) + C \).

Step-by-step solution

Identify the Substitution

To solve the integral \( \int \frac{e^{2x}}{1+e^{4x}} \, dx \), we first look for an appropriate substitution. Notice that the expression \( e^{4x} \) appears in the denominator. We can use the substitution \( u = e^{2x} \), which simplifies the computations.

Differentiate the Substitution

After choosing \( u = e^{2x} \), we differentiate it to link it to \( dx \). The derivative is \( \frac{du}{dx} = 2e^{2x} \), or rearranging gives \( du = 2e^{2x} \, dx \). Thus, \( e^{2x} \, dx = \frac{1}{2} du \).

Rewrite the Integral in Terms of 'u'

Substitute \( u = e^{2x} \) and \( e^{2x} \, dx = \frac{1}{2} du \) into the integral. This changes the integral to \( \int \frac{1}{1+u^2} \cdot \frac{1}{2} du \), which simplifies to \( \frac{1}{2} \int \frac{1}{1+u^2} du \).

Find the Antiderivative

The integral \( \int \frac{1}{1+u^2} du \) is a standard form and its antiderivative is known to be \( \arctan(u) + C \), where \( C \) is the constant of integration. Therefore, \( \frac{1}{2} \int \frac{1}{1+u^2} du = \frac{1}{2} \arctan(u) + C \).

Substitute Back to Original Variable

Finally, substitute back \( u = e^{2x} \) into the antiderivative. This results in \( \frac{1}{2} \arctan(e^{2x}) + C \).

Key concepts

Substitution Method

The Substitution Method is a technique in integral calculus used to simplify finding antiderivatives. The main idea involves changing variables to transform a complicated integral into a simpler one. Here's how it generally works:

• Choose a substitution: You identify part of the integral's expression, which can be replaced by a single variable, say, \( u \). This step simplifies the integral's structure.
• Differentiate the substitution: Find \( \frac{du}{dx} \), the derivative of your substitution equation, to later replace \( dx \) in the integral.
• Rewrite the integral: Substitute all instances of the chosen part and \( dx \) in your integral. This converts it into an integral in terms of \( u \).
• Solve the new integral: Often, the new integral will be simpler or be in a standard form that is easier to solve.
• Substitute back: After finding the antiderivative in terms of \( u \), revert it back to the original variable, completing the solution.

This method is particularly handy when the integral involves polynomial expressions, trigonometric identities, or exponential functions, which are difficult to integrate otherwise. In our exercise, by choosing \( u = e^{2x} \), the integration becomes straightforward.

Integral Calculus

Integral Calculus is a fundamental branch of mathematics focused on accumulation and the area under curves. It complements differential calculus, offering methods to deduce information about functions that describe quantities accumulating over time. Key ideas include:

• The Integral: An integral sums infinitesimally small quantities over a region — often an interval on the real number line.
• Antiderivatives: If \( F(x) \) is an antiderivative of \( f(x) \), the process of integration finds such functions \( F \). This means that \( \frac{d}{dx}[F(x)] = f(x) \).
• The Indefinite Integral: Represented as \( \int f(x) \, dx \), it denotes the set of all antiderivatives of \( f(x) \) along with a constant of integration \( C \).
• Definite vs. Indefinite Integrals: A definite integral \( \int_a^b f(x) \, dx \) computes the exact accumulation between bounds \( a \) and \( b \); an indefinite integral remains general.

Understanding these concepts helps you solve various problems, ranging from computing areas to modeling natural phenomena. The exercise at hand, finding the antiderivative of the given function, taps directly into these foundational principles.

Trigonometric Integration

Trigonometric Integration is a method specifically designed for integrating functions involving trigonometric functions such as sine, cosine, and tangent. It encompasses techniques like:

• Standard integrals: Many trigonometric integrals have standard solutions, like \( \int \sin(x) \, dx = -\cos(x) + C \), \( \int \cos(x) \, dx = \sin(x) + C \), etc.
• Trigonometric identities: Using identities, such as \( \sin^2(x) + \cos^2(x) = 1 \), to simplify integrals is vital — they can convert complex expressions into simpler ones.
• Inverse trigonometric functions: Often, when a substitution reduces an integral to a form with \( \arctan \), \( \arcsin \), or \( \arccos \), understanding these functions and their derivatives becomes critical.
• Reduction formulas: Some integrals are tackled by systematically reducing their degree using specific formulas, which help solve integral powers of trigonometric functions.

In the original solution, the integral \( \int \frac{1}{1+u^2} du \) directly ties into this concept because of its standard solution linked to inverse trigonometric functions — here, the \( \arctan \) function.