Problem 43 - Calculus and its applications

Question

Evaluate. $$ \int_{0}^{1} 2 x e^{x^{2}} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( e - 1 \).

Step 1: Identify the type of integral

First, recognize that we have an integral of the form \( \int 2x e^{x^2} \, dx \). This suggests the use of substitution, as the integrand includes a composite function.

Step 2: Choose an appropriate substitution

Let's set \( u = x^2 \). Then, we find the differential \( du = 2x \, dx \). This choice simplifies the integral, as \( 2x \, dx \) is already part of the integrand.

Step 3: Change the limits of integration

When \( x = 0 \), \( u = 0^2 = 0 \). When \( x = 1 \), \( u = 1^2 = 1 \). Thus, the definite integral's limits also change according to this substitution, from \( x = 0 \) to \( x = 1 \) into \( u = 0 \) to \( u = 1 \).

Step 4: Rewrite the integral with substitution

Replace parts in the integral using substitution: the integral \( \int_{0}^{1} 2x e^{x^2} \, dx \) becomes \( \int_{0}^{1} e^u \, du \).

Step 5: Evaluate the integral

The integral \( \int e^u \, du \) is equal to \( e^u + C \). Evaluating from \( u = 0 \) to \( u = 1 \), we get:\[ e^1 - e^0 = e - 1 \]

Step 6: Conclude the evaluation

The evaluated integral is \( e - 1 \). This is the final result of the original integral.

Key Concepts

Substitution Method

The substitution method is a powerful technique in integral calculus. It is especially useful when dealing with composite functions. The idea is to simplify the integral by changing variables. In our original exercise, the integral \( \int_{0}^{1} 2x e^{x^2} \, dx \) involves a composite function \( e^{x^2} \). By defining a new variable \( u = x^2 \), we transform the integral into a simpler form. To complete this substitution, we also need to express \( dx \) in terms of \( du \). Calculating the derivative gives us \( du = 2x \, dx \), which perfectly matches part of our integrand. This strategically chosen substitution transforms the integral into \( \int e^u \, du \), an expression we can now integrate easily.

Definite Integral

A definite integral, unlike its ambiguous counterpart, the indefinite integral, is all about finding the exact area under a curve between two points. In this problem, we calculate the integral from \( x = 0 \) to \( x = 1 \). When performing substitution, it's crucial to also change the integration limits to match the new variable. Initially, our limits of integration were expressed in terms of \( x \):

• When \( x = 0 \), \( u = 0^2 = 0 \).
• When \( x = 1 \), \( u = 1^2 = 1 \).

Now, our new limits are from \( u = 0 \) to \( u = 1 \). This ensures the evaluation remains consistent with the transformed function \( e^u \). The evaluated integral, \( int e^u \, du \), reveals the precise area under the curve which translates to \( e - 1 \).

Exponential Function

The exponential function \( e^x \) is one of the most important functions in mathematics, characterized by its constant rate of growth. In the context of this integral, our focus is on \( e^{x^2} \), a form that signifies rapid growth due to the square in the exponent. Exponential functions are unique because when you integrate an exponential function like \( e^u \), the result still involves the exponential function:

• The integral of \( e^u \) is \( e^u + C \), where \( C \) is the constant of integration for indefinite integrals.

When working with definite integrals, we calculate the difference at the endpoints of our limits, leading to our final results. For example, the simple subtraction of expressions like \( e^1 - e^0 \) results in \( e - 1 \), highlighting the exponential function's contribution to the area under the curve.