Question

Evaluate the following integrals. Include absolute values only when needed. $$\int_{-2}^{2} \frac{e^{z / 2}}{e^{z / 2}+1} d z$$

Short answer

Question: Calculate the definite integral of the function $$\frac{e^{z / 2}}{e^{z / 2}+1}$$ with respect to z, from -2 to 2. Answer: $$2\ln \frac{1+\frac{1}{\sqrt{e}}}{1-\frac{1}{\sqrt{e}}}$$

Step-by-step solution

Identify and rewrite the function

We first need to identify and rewrite the given function as follows in order to ease its integration: $$\int_{-2}^{2} \frac{e^{z / 2}}{e^{z / 2}+1} d z$$ Now let's proceed to the next step.

Perform the integration using substitution

We perform integration by substitution. Let's consider a variable 't' which satisfies the following equation: $$t = e^{\frac{z}{2}} \Rightarrow \frac{dt}{dz} = \frac{1}{2}e^{\frac{z}{2}}$$ Now we can rewrite the integrand in terms of 't': $$\int_{-2}^{2} \frac{e^{z / 2}}{e^{z / 2}+1} d z = 2\int_{t(-2)}^{t(2)} \frac{1}{t+1} dt$$ The next step is to evaluate this integral.

Evaluate the definite integral within the given limits

The integral of the function $$\frac{1}{t+1}$$ is: $$\int \frac{1}{t+1} dt = \ln |t+1| \Rightarrow 2\int \frac{1}{t+1} dt = 2\ln |t+1| + C$$ Now we need to calculate the definite integral within the bounds t(-2) and t(2): $$2\ln |t(\frac{1}{\sqrt{e}})+1| - 2\ln |t(-\frac{1}{\sqrt{e}})+1| = 2\ln \frac{1+\frac{1}{\sqrt{e}}}{1-\frac{1}{\sqrt{e}}}$$ The final result of evaluating the integral is: $$\int_{-2}^{2} \frac{e^{z / 2}}{e^{z / 2}+1} d z = 2\ln \frac{1+\frac{1}{\sqrt{e}}}{1-\frac{1}{\sqrt{e}}}$$

Key concepts

Definite Integrals

Definite integrals are a fundamental tool in calculus used to find the area under a curve within a specified interval. Unlike indefinite integrals, which yield a general form or family of functions, definite integrals provide a specific numerical value. This value represents the accumulation of quantities, such as area or total change, over a range from a lower limit to an upper limit.
The function to be integrated is expressed with its boundaries, often written as \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) are the lower and upper limits, respectively. One of the key properties of definite integrals is their dependency on the Fundamental Theorem of Calculus, which bridges the concept of differentiation and integration.
When evaluating definite integrals, it’s important to interpret the physical significance of the problem correctly. In our exercise, we are interested in the actual value of the integral from \(-2\) to \(2\), which requires careful consideration of how the function behaves across these limits.

Integration Techniques

Integration techniques are methods used to find the integral of functions, especially when they aren't straightforward to solve. There are several approaches, each suitable for different types of problems. For basic functions, straightforward integration rules can apply, like power and constant rules.
However, more complex functions often require advanced techniques such as:

• Substitution
• Integration by Parts
• Partial Fractions
• Trigonometric Substitutions

These methods can simplify the problem into more manageable parts, or transform it into a form where standard integration rules can be applied.
In our example, the integrand \( \frac{e^{z / 2}}{e^{z / 2}+1} \) suggests a substitution method due to the presence of exponential functions, which allows for a simpler integration process post-substitution.

Substitution Method

The substitution method, often likened to the reverse of the chain rule in differentiation, is a key integration technique for simplifying complex integral forms. It methodically replaces a difficult expression with a simpler variable, transforming the integral into a form that is easier to solve.
In this method, a suitable substitution is chosen. For our example, we set \( t = e^{z/2} \). Consequently, the differential \( dt = \frac{1}{2}e^{z/2} \, dz \), allowing us to rewrite and simplify the integrand.
Performing this substitution transforms the original integral into a simpler one: \( 2\int_{t(-2)}^{t(2)} \frac{1}{t+1} \, dt \). This reformulated integral relates directly to the natural logarithm, yielding \( 2\ln|t+1| \) upon solving.
Understanding when and how to use substitution can make integrating complex functions more manageable, which is crucial in both mathematical theory and practical applications.