Problem 51 - Calculus

Question

Evaluate the integrals. $$ \int_{0}^{1} e^{1+x} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( e^2 - e \).

Step 1: Understand the Integral

The given integral is \( \int_{0}^{1} e^{1+x} \, dx \). This is a definite integral with limits from 0 to 1. The function to integrate is \( e^{1+x} \).

Step 2: Simplify the Integrand

Notice that \( e^{1+x} = e \cdot e^x \). The constant \( e \) can be factored out of the integral. Thus, the integral simplifies to \( e \int_{0}^{1} e^x \, dx \).

Step 3: Integrate the Exponential Function

Find the indefinite integral of \( e^x \), which is \( e^x \). So the integral becomes \( e \cdot e^x + C \) where \( C \) is the constant of integration for the indefinite integral.

Step 4: Evaluate the Definite Integral

Since the integral is definite from 0 to 1, evaluate \( e^x \) at these bounds: \( e \cdot (e^1 - e^0) = e \cdot (e - 1) \).

Step 5: Compute the Answer

Calculate the result by completing the calculation \( e(e - 1) \). This results in \( e^2 - e \).

Key Concepts

Definite Integrals

Definite integrals are a fundamental concept in calculus, often representing the area under a curve between specific points on the x-axis. They allow you to calculate the net change of a function over a given interval. In our exercise, the definite integral is marked by the integral sign with limits of integration from 0 to 1 for the function \( e^{1+x} \).
To evaluate a definite integral, you first find the antiderivative of the integrand, but unlike an indefinite integral, you apply the limits of integration after finding the antiderivative. This changes it from a general formula into a specific numeric answer.

• Lower and upper limits: Indicated as subscript and superscript on the integral sign.
• The Fundamental Theorem of Calculus: This theorem links the process of differentiation and integration, allowing you to solve definite integrals.
• Calculating the result: Evaluate the antiderivative at the upper limit and subtract the evaluation at the lower limit.

These steps result in specific values giving a concrete output unlike the general formula obtained with indefinite integrals.

Exponential Functions

Exponential functions have the form \( a^x \) or more commonly, \( e^x \), where \( e \) is the natural logarithm base, approximately equal to 2.71828. These functions model continuous growth or decay processes and appear frequently in various fields of science and mathematics.
In the context of integration, exponential functions are particularly 'friendly' because the derivative and the integral of \( e^x \) remain \( e^x \). This property makes integrating functions with exponential forms straightforward.

• Model for Growth/Decay: Used in problems involving population growth, radioactive decay, and more.
• Properties: The function \( e^x \) exhibits rapid growth, doubling for every increase by a small constant.
• Integration: Direct and follows the same form as its derivative, simplifying calculations.
• Shifts: A function like \( e^{1+x} \) can be simplified using exponent rules: \( e \cdot e^x \).

Exponential functions in integrals often simplify the process, reducing the complexity of calculations.

Integration Techniques

Integration techniques are methods applied to find the integral of a function. Various techniques are used depending on the function you need to integrate. These methods make integrating complex functions more approachable.
Basic techniques include,factoring constants from the integral sign and recognizing patterns that match standard integrals.

• Factoring Constants: As seen in the exercise, factor constants out of the integrand to simplify calculations.
• Standard Integrals: Recognize basic forms like \( \int e^x \, dx = e^x + C \).
• Advanced Techniques: For more complex functions, techniques like integration by parts, partial fraction decomposition, and substitution are used.
• Simplification: Always look for ways to simplify the integrand, as in \( e^{1+x} = e \, e^x \).

Choosing the right integration technique helps in tackling even complex integrals easier, leading to more efficient problem solving in calculus.