Question

Evaluate the following integrals. $$\int \frac{e^{2 z}}{e^{2 t}-4 e^{-z}} d z$$

Short answer

Question: Evaluate the indefinite integral \(\int \frac{e^{2z}}{e^{2t}-4e^{-z}} dz\). Answer: The indefinite integral evaluates to \(\frac{1}{2} \ln |e^z-2 e^{t}| + \frac{1}{2} \ln |e^z +2 e^{t}| + C\), where C is the constant of integration.

Step-by-step solution

Simplify the integrand

To make the integration process easier, let's simplify the integrand first. Since the exponential functions have the same base, we can manipulate them using the property \(e^a / e^b = e^{a-b}\). So, the integrand becomes: $$\frac{e^{2 z}}{e^{2 t}-4 e^{-z}} = \frac{e^{2 z}(e^{z})^2}{(e^{2 t})(e^{z})^2-4} = \frac{e^{3 z}}{e^{2 t+z}-4}$$

Apply substitution

Now, we will use the substitution method to make the integration easier. Let \(u = e^{z}\), then, \(z = \ln u\), and \(\frac{du}{dz} = e^z = u\). Converting \(dz\) to \(du\), we get: $$dz = \frac{1}{u} du$$ Now, let's substitute these variables in our integral, simplifying fractions: $$\int \frac{e^{3 z}}{e^{2 t+z}-4} dz = \int \frac{u^{3} u}{u^{2} e^{2 t} - 4} \frac{1}{u} du = \int \frac{u^2}{u^{2} e^{2 t} - 4} du$$

Partial fraction decomposition

To integrate the obtained fraction, we will use partial fraction decomposition. We want to rewrite our fraction in the form: $$\frac{u^2}{u^{2} e^{2 t} - 4} = \frac{A}{u - 2 e^{t}} + \frac{B}{u + 2 e^{t}}$$ Multiplying both sides by \((u^{2} e^{2 t} - 4)\), we get: $$u^2 = A(u + 2 e^{t}) + B(u - 2 e^{t})$$ By equating the coefficients of \(u^2\), \(u^1\), and \(u^0\), we get \(A=1/2\) and \(B=1/2\). So, our fraction becomes: $$\frac{u^2}{u^{2} e^{2 t} - 4} = \frac{1/2}{u-2 e^{t}} + \frac{1/2}{u+ 2 e^{t}}$$

Integrate

Now, we just need to integrate each term: $$\int \frac{u^2}{u^{2} e^{2 t} - 4} du = \int \left(\frac{1/2}{u-2 e^{t}} + \frac{1/2}{u+ 2 e^{t}}\right) du = \frac{1}{2} \int \frac{1}{u-2 e^{t}} du + \frac{1}{2} \int \frac{1}{u+ 2 e^{t}} du$$ These integrals are straightforward. Their antiderivatives are natural logarithms: $$\frac{1}{2} \ln |u-2 e^{t}| + \frac{1}{2} \ln |u+ 2 e^{t}| + C$$

Substitute back

Finally, substitute \(u = e^z\) back into the expression to get the final result: $$\frac{1}{2} \ln |e^z-2 e^{t}| + \frac{1}{2} \ln |e^z +2 e^{t}| + C$$

Key concepts

Substitution Method

Integration can sometimes be tricky, but the substitution method simplifies our task significantly. It's almost like magic to transform complex integrals into simpler forms by changing variables. In this method, we choose a new variable for the integral that makes our calculations easier. For instance, in our exercise, we let \( u = e^z \). This decision is strategic because it simplifies the exponential terms.

By substituting \( z = \ln u \), we relate the new variable to the old one using logarithms. The differential \( dz \) also transforms using the chain rule: \( dz = \frac{1}{u} du \), ensuring all terms in the integral are comparable with respect to \( du \). This setup transforms the integral \( \int \frac{e^{3z}}{e^{2t+z}-4} dz \) into a much simpler form \( \int \frac{u^2}{u^{2} e^{2t} - 4} du \).

The substitution method is a powerful tool for simplifying integrals to a point where other techniques, such as partial fraction decomposition, can be effectively used.

Partial Fraction Decomposition

Once the integral is simplified using substitution, you might encounter a rational function. That's where partial fraction decomposition shines. It breaks down a complex rational expression into simpler fractions, which are easier to integrate individually.

For this exercise, we decomposed \( \frac{u^2}{u^{2} e^{2t} - 4} \) into simpler terms. These fractions are expressed as \( \frac{A}{u - 2 e^t} + \frac{B}{u + 2 e^t} \). By equating coefficients of corresponding powers of \( u \), we solve for these constants: \( A = \frac{1}{2} \) and \( B = \frac{1}{2} \).

This decomposition is crucial as it turns a difficult integral into a sum of simpler integrals, each involving terms you can handle with basic integration techniques. It's like untangling a complex knot into simpler threads you can manage one at a time.

Natural Logarithm

The natural logarithm is a fundamental concept in calculus and appears often in integration, especially when dealing with exponential functions. It connects deeply with the derivative and integral of \( \, e^x \, \), the base of the natural logarithms.

In our case, after using partial fraction decomposition, our integral transformed into simpler terms. Each of these terms, like \( \frac{1}{u - 2 e^t} \), can be integrated using the natural logarithm. The integral \( \int \frac{1}{x} dx \) is \( \ln |x| + C \). This property allows us to write the integral of these decomposed terms as natural logarithms: \( \frac{1}{2} \ln |u-2 e^t| \) and \( \frac{1}{2} \ln |u+ 2 e^t| \).

Natural logarithms simplify the representation and understanding of integrals, particularly when dealing with expressions that resolve to a form \( \frac{1}{x} \). It is an elegant function that seamlessly connects exponential and logarithmic ideas.

Antiderivatives

Antiderivatives are the reverse process of differentiation and form the essence of solving integrals. When asked to find an antiderivative, you are essentially finding a function whose derivative is the original integrand. The resulting expression is also known as the indefinite integral, which includes an integration constant \( C \).

In our exercise, after performing all necessary steps, including substitution and decomposition, we integrated straight forward expressions. Each expression gave a natural logarithm as its antiderivative. Our final expression \( \frac{1}{2} \ln |e^z-2 e^t| + \frac{1}{2} \ln |e^z +2 e^t| + C \) is the antiderivative of the transformed integral.

Understanding antiderivatives involves recognizing how to reverse differential operations. It's an essential skill in calculus, helping you to determine original functions based on their rates of change. The focus is not only on finding these functions but on understanding their behavior and implications in the context of integration.