Question

In the following exercises, compute each integral using appropriate substitutions. \(\int \frac{e^{t}}{1+e^{2 t}} d t\)

Short answer

The integral evaluates to \(\arctan(e^t) + C\).

Step-by-step solution

Identify an Appropriate Substitution

To simplify the integral, we look for a substitution that will make the integrand easier to integrate. Notice that the derivative of the denominator \(1+e^{2t}\) is related to \(e^{2t}\). Hence, the substitution \(u = e^{t}\) might work, as \(e^{2t} = u^2\).

Differentiate Substitution

If \(u = e^{t}\), then \(du = e^{t} dt\). This allows us to express \(dt\) in terms of \(du\): \[ dt = \frac{du}{u} \]

Substitute into the Integral

Substitute \(u = e^{t}\) and \(dt = \frac{du}{u}\) into the integral:\[ \int \frac{e^{t}}{1+e^{2t}} dt = \int \frac{u}{1+u^2} \cdot \frac{du}{u} = \int \frac{1}{1+u^2} du \]

Simplify and Integrate

Now integrate \(\int \frac{1}{1+u^2} du\), which is a known result:\[ \int \frac{1}{1+u^2} du = \arctan(u) + C \]

Substitute Back to Original Variable

As the integration is done, substitute back \(u = e^{t}\) to express the integration in terms of \(t\):\[ \arctan(u) + C = \arctan(e^{t}) + C \]

Final Answer

The evaluated integral is:\[ \int \frac{e^{t}}{1+e^{2t}} dt = \arctan(e^{t}) + C \]

Key concepts

Substitution Method

The substitution method is a powerful tool for solving integrals, especially when the integrand involves composite functions.
The idea is to change variables in such a way that the new integral is easier to solve. In our case, we used this method effectively to simplify the given integral.When you make a substitution, you're looking to rewrite part of the integral in terms of a new variable, often "u."
In our example, we had the expression \(\int \frac{e^{t}}{1+e^{2t}} dt\).
Recognizing that the derivative of the denominator \(1+e^{2t}\) is closely related to \(e^{2t}\), we chose \(u = e^{t}\). This choice was strategic because it allows us to transform the denominator and the numerator simultaneously, making the integration process more straightforward.Here's a tip:- Make sure your substitution simplifies the integral significantly. This method is most effective when it turns a complicated integral into a basic form you can easily handle.

Exponential Functions

Exponential functions, like \(e^{t}\), play a significant role in calculus due to their unique properties.
These functions are used in various applications, including growth and decay models.In integration, dealing with exponential terms often leads us to consider how they relate to other parts of a function.
In our problem, \(e^{t}\) not only formed the numerator but was also transformed into the denominator \(e^{2t}\) via squaring.
Recognizing this transformation was key in making our substitution.To handle exponential functions in integration:- Familiarize yourself with derivatives and integrals of exponential functions.- Look for opportunities where exponentials can replace entire sub-expressions or simplify the form.Keep in mind how these functions behave under addition, multiplication, and especially within calculus operations.

Trigonometric Integration

Trigonometric integration often involves using identities or inverses of trigonometric functions to solve integrals.
These are critical when dealing with integrals that contain or transform into trigonometric forms.Our integral eventually came to involve the term \(\frac{1}{1+u^2}\), which is recognizable as the derivative of \(\arctan(u)\).
This is a classic result in trigonometric integration; knowing it allows us to quickly integrate to find \(\int \frac{1}{1+u^2} du = \arctan(u) + C\).For efficient trigonometric integration:- Keep handy common derivatives and integrals related to trigonometric expressions.- Learn the basic trigonometric identities and how they can transform integrals.With practice, spotting these transformations becomes intuitive, streamlining the process of integration.