Problem 208 - Calculus Volume 1

Question

Use basic integration formulas to compute the following antiderivatives. $$ \int\left(e^{2 x}-\frac{1}{2} e^{x / 2}\right) d x $$

Step-by-Step Solution

Answer

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

Step 1: Break Down the Integral

We have the integral \( \int \left( e^{2x} - \frac{1}{2} e^{x/2} \right) dx \). Notice that it's a combination of two functions: \( e^{2x} \) and \( -\frac{1}{2} e^{x/2} \). The integral of a sum or difference of functions is the sum or difference of their integrals.

Step 2: Integrate Each Term Separately

Start by integrating \( e^{2x} \). The formula for the integral of \( e^{ax} \) is \( \frac{1}{a} e^{ax} + C \). Therefore, \( \int e^{2x} \, dx = \frac{1}{2} e^{2x} + C_1 \).

Step 3: Integrate the Second Term

Now integrate \( -\frac{1}{2} e^{x/2} \). The formula for this is similar where \( a = \frac{1}{2} \). So, \( \int -\frac{1}{2} e^{x/2} \, dx = -\frac{1}{2} \times \frac{1}{1/2} e^{x/2} + C_2 = -e^{x/2} + C_2 \).

Step 4: Combine the Results

Combine both integrals: \( \int \left( e^{2x} - \frac{1}{2} e^{x/2} \right) dx = \frac{1}{2} e^{2x} - e^{x/2} + C \), where \( C \) represents the overall constant of integration (\( C = C_1 + C_2 \)).

Key Concepts

Antiderivatives

An antiderivative is essentially the reverse of taking a derivative. It is a function that, when differentiated, gives back the original function you started with. This process is also known as integration. Think of it as "un-differentiating" a function.

When you are tasked with finding an antiderivative, you need to determine a function whose derivative is the given function. This concept is important because it helps us solve problems related to area under curves and in various applied fields like physics and engineering.

In practical terms, finding antiderivatives involves using integration techniques. These techniques can be as straightforward as reversing the rules of differentiation, or more complex, requiring special methods like substitution or integration by parts. Mastering antiderivatives involves recognizing the type of function you're working with and applying the right technique to "undo" the derivative.

Basic Integration Formulas

Basic integration formulas are like a toolkit for finding antiderivatives. They provide simple rules that help you integrate common functions with ease. These rules are essential for solving a wide variety of integration problems.

Some key formulas include:

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) for \( n eq -1 \)
• \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \)
• \( \int \frac{1}{x} \, dx = \ln|x| + C \)

Understanding these basic formulas allows you to quickly recognize integral patterns and apply the appropriate formula to solve them. They also save you time when dealing with more complex integrals because you can break down larger problems into smaller, manageable parts that match these formulas.

Keep in mind that each formula includes a \(+ C\), which is the constant of integration. This shows that there are infinitely many antiderivatives for any given function, differing only by a constant.

Exponential Functions Integration

Integration of exponential functions is a vital skill since exponential functions frequently appear in various mathematical problems and real-world applications. The most common type of exponential integral involves the function \(e^x\).

An important integration formula to remember is \(\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C\). This formula is crucial when working with exponential functions since it allows us to find their antiderivatives directly.

In the context of integrating exponential terms in exercises, you often need to handle functions like \(e^{2x}\) or \(e^{x/2}\). For these, the constant \(a\) in the exponent affects the integration, as seen in:\

• \(\int e^{2x} \, dx = \frac{1}{2} e^{2x} + C\)
• \(\int -\frac{1}{2} e^{x/2} \, dx = -e^{x/2} + C\)

Each requires careful handling of the rate of growth or decay represented by the exponent, remembering to adjust the coefficient accordingly. Mastery of exponential integration concepts allows you to tackle both simple and complex exponential integrals with confidence.