Problem 45 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{0}^{1} x e^{x^{2}} d x\)

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{1}{2}(e - 1) \).

Step 1: Identify the Substitution

The integral is \( \int_{0}^{1} x e^{x^{2}} \ dx \). We will use the substitution method. Consider the substitution \( u = x^2 \). Then, \( du = 2x \, dx \), or equivalently \( dx = \frac{du}{2x} \).

Step 2: Solve for dx and Replace

From our substitution, we have \( dx = \frac{du}{2x} \). Substitute \( u = x^2 \) and \( dx = \frac{du}{2x} \) into the integral: \( \int x e^{u} \cdot \frac{du}{2x} \). The \( x \) terms cancel, reducing the expression to \( \frac{1}{2} \int e^u \, du \).

Step 3: Determine New Limits of Integration

Since our original integral had limits from 0 to 1 for \( x \), we need to find corresponding limits for \( u \). When \( x = 0 \), \( u = 0^2 = 0 \). When \( x = 1 \), \( u = 1^2 = 1 \). Hence, the new limits for \( u \) are also from 0 to 1.

Step 4: Evaluate the Integral

Our integral has simplified to \( \frac{1}{2} \int_{0}^{1} e^u \, du \). The antiderivative of \( e^u \) is \( e^u \). Thus, \[ \frac{1}{2} \left[ e^u \right]_{0}^{1} = \frac{1}{2} \left( e^{1} - e^{0} \right). \]

Step 5: Simplify the Result

Now evaluate \( \frac{1}{2} \left( e - 1 \right) \). Since \( e^0 = 1 \), this simplifies to \( \frac{1}{2} (e - 1) \).

Key Concepts

Definite Integrals

Definite integrals are a fundamental concept in calculus. They help us calculate the net area under a curve within specified limits, providing us with a numerical value. In our problem, \[ \int_{0}^{1} x e^{x^{2}} \ dx \]is a definite integral from 0 to 1. This means we are finding the area under the curve of the function \( x e^{x^{2}} \) between \( x = 0 \) and \( x = 1 \) on the x-axis.
Unlike indefinite integrals, which result in a family of functions, definite integrals yield specific values due to their fixed bounds. It is crucial to evaluate the function within these bounds to obtain the precise result.

In short, definite integrals give us valuable insights into quantities represented by a curve over a specific interval. They play a significant role in various scientific and engineering applications where we need precise measurements.

Substitution Rule

The substitution rule is a powerful technique in calculus used to simplify the integration process. It involves replacing a part of the integrand with a new variable that makes the integral easier to solve. For the given integral, we use the substitution:

• Let \( u = x^2 \)
• This implies \( du = 2x \, dx \) or \( dx = \frac{du}{2x} \)

This substitution transforms our problem into a simpler form. By changing variables, complicated integrations often become straightforward, leading to easier solutions.

Switching from \( x \) to \( u \), we simplify the integral \[ \int_{0}^{1} x e^{x^{2}} \ dx \]into \[ \frac{1}{2} \int e^u \, du \].Such substitution effectively handles tricky integrals that are otherwise difficult to integrate directly.

Integration Techniques

Several integration techniques help tackle various integrals in calculus. In this problem, using the substitution method, we simplified the complex integral to an easily solvable one. Techniques commonly used besides substitution include:

• Integration by Parts
• Partial Fraction Decomposition
• Trigonometric Substitutions

Each technique has its unique application and is selected based on the form of the integral at hand. Often, combining multiple techniques can be necessary to solve intricate problems.

The effectiveness of integration techniques comes from their ability to transform difficult-to-integrate expressions into manageable forms. This power makes calculus approachable and applicable in real-world scenarios. Always choose the technique that best fits the integral for efficient computation.

Limit of Integration

The limits of integration mark the start and end points of a definite integral, directly affecting the calculated area. For the problem \[ \int_{0}^{1} x e^{x^{2}} \ dx \], the limits are initially in terms of \( x \), ranging from 0 to 1.

When using the substitution rule, it's necessary to convert these limits to the new variable's terms. Here, upon substitution \( u = x^2 \), and calculating for upper and lower bounds, we find:

• When \( x = 0 \), \( u = 0 \)
• When \( x = 1 \), \( u = 1 \)

Thus, the limits for \( u \) also occur from 0 to 1.
The correction ensures that the integral is evaluated correctly in its new form. Misalignment in limits can lead to errors in results, emphasizing their importance in calculus problems.