Problem 8 - College Algebra and Calculus: An Applied Approach

Question

Use the Exponential Rule to find the indefinite integral. $$ \int(2 x+1) e^{x^{2}+x} d x $$

Step-by-Step Solution

Answer

\(e^{x^{2} + x} + C\)

Step 1: Identify the function and its derivative

First, we need to identify a function and its derivative within the integrand. We can see that the function \(f(x) = e^{x^2 + x}\) is multiplied by its derivative \(f'(x) = (2x + 1)e^{x^2 + x}\). This means we can use the rule of integration to simplify the integral.

Step 2: Apply the Exponential Rule

We use the Exponential Rule here: ∫f'(x)*e^(f(x)) dx = e^(f(x)), where f(x) = \(x^2 + x\). By this rule, our integral simplifies to \( e^{x^2 + x} + C \), where \(C\) is the constant of integration.

Key Concepts

Exponential Rule

The Exponential Rule for integration refers to a specific technique used in calculus that simplifies the process of integrating functions of the form \(e^{f(x)}\) multiplied by their derivatives. The rule is based on the relationship between the exponential function and its derivative, which remains proportional to the original function. In essence, when you spot a function times its derivative—typically involving an exponential term—you can apply the Exponential Rule.

For example, consider the integral \[\int(2x+1)e^{x^{2}+x}dx\]. The term \(2x+1\) is the derivative of the function \(x^2+x\), and the entire integrand has the form of a function times the derivative of an exponential function. By applying the Exponential Rule, we can directly integrate to obtain \[e^{x^2 + x} + C\], significantly simplifying the integration process. The constant \(C\) symbolizes the arbitrary constant that is inherent to indefinite integrals.

integration technique

Integration techniques are various methods used to evaluate integrals in calculus. There are several common techniques, including substitution, integration by parts, trigonometric substitution, partial fractions, and the Exponential Rule, among others. Each technique addresses a specific form or structure of an integrand.

When a problem involves an integrand that is the product of a function and its derivative, especially within an exponential function, using a straightforward integration technique like the Exponential Rule becomes optimal. This approach allows for direct integration without the need for more complex or lengthier methods, granting a clear pathway to the solution. Understanding when and how to effectively apply an integration technique is vital for students of calculus—it is the key to solving integrals with precision and efficiency.

calculus

Calculus is a branch of mathematics that studies how things change. It's split primarily into two areas: differential calculus, concerned with the concept of a derivative, and integral calculus, focusing on the concept of an integral. Calculus is fundamentally used in various fields, including science, engineering, economics, and statistics, to model and solve problems involving dynamic systems or processes.

In the context of our example, the indefinite integral \[\int(2x+1)e^{x^{2}+x}dx\] exemplifies how calculus can be used to find the antiderivative of complex functions. By understanding the foundational rules and techniques, such as the Exponential Rule, students are empowered to tackle a wide array of problems by breaking them down into simpler forms, ultimately leading to a greater understanding of the behavior of functions over an interval.