Problem 18 - Calculus: Early Transcendental Functions

Question

Find the general antiderivative. $$\int\left(4 x-2 e^{x}\right) d x$$

Step-by-Step Solution

Answer

The general antiderivative of the given function is \(2x^2 - 2e^{x} + C\).

Step 1: Identify the Terms

The function to be integrated can be broken down into two terms: \(4x\) and \(-2e^{x}\). Each of these terms will be integrated separately.

Step 2: Apply the Integral

For the first term, the power rule of integration is used, i.e., the integral of \(x^n\) is \(\frac{1}{n+1}x^{n+1}\). Here \(n = 1\), so this results in \(\frac{1}{1+1}x^{1+1} = 2x^2\). For the second term, the integral of \(e^x\) is \(e^x\), so the integral of \(-2e^x\) is \(-2e^x\). The minus sign and coefficient of 2 are preserved as the antiderivative of a constant times a function is equal to that constant times the antiderivative of the function.

Step 3: Add the Integration Constant

When finding the indefinite integral, it is crucial to include the constant of integration, often denoted as +C. This is because the derivative of any constant is zero, thus without loss of generality, the general antiderivative is expressed.

Key Concepts

Indefinite Integrals

Indefinite integrals are a foundational concept in calculus. They deal with finding the antiderivative of a function, which essentially "undoes" differentiation.

• This involves finding a function whose derivative matches the original function you started with.
• The result is a family of functions, hence the term "indefinite" integral implying a broad range.
• Indefinite integrals employ the integral symbol \( \int \), but unlike definite integrals, they do not have specified upper and lower limits.

To illustrate this with an example, consider the function \( f(x) = 4x - 2e^x \). The indefinite integral or the general antiderivative is found by integrating each term separately. When computing indefinite integrals, it's essential to not forget the constant of integration denoted \(+C\). This constant represents all vertically shifted versions of a potential antiderivative that still differentiate back to the original function.By integrating term-by-term, you find the family of functions that could potentially be derived back to the original function.

Power Rule

The power rule is a simple and powerful tool used in calculus to find antiderivatives for polynomial expressions. It is fundamental for solving problems involving indefinite integrals.

• The power rule states that the antiderivative of \(x^n\) is \(\frac{1}{n+1}x^{n+1}\), provided \(n eq -1\).
• This rule is derived from the basic principle of reversing differentiation.

For instance, in our initial problem involving \(4x\), \(n\) is 1. Applying the power rule: \[\int 4x\, dx = 4 \cdot \frac{1}{2}x^2 = 2x^2\]The power rule simplifies the process of finding antiderivatives by providing a straightforward method to transform polynomials.It's crucial to manage the coefficient correctly, multiplying it after applying the rule, ensuring each term is handled precisely and efficiently.

Exponential Functions

Exponential functions have a distinctly different behavior in calculus compared to polynomial expressions. They often appear in real-world problems involving growth and decay.

• An exponential function is characterized by the constant \(e\), approximately 2.718, which is the base of natural logarithms.
• The integral of an exponential function \(e^x\) is straightforward: it is \(e^x\) again due to its unique property that its rate of change is proportional to its current value.

For the term \(-2e^x\) in our exercise, this pattern applies:\[\int -2e^x\, dx = -2e^x\]Thus, the antiderivative of an exponential function retains its exponential form. It's just adjusted by constants as they appear in front of the \(e^x\) term, without altering its fundamental growth property.Don't forget to add the integration constant \(+C\) at the end to express the general solution of the indefinite integral.