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Chapter 13: Problem 31

Evaluate the integrals. $$ \int_{0}^{1} 5 x e^{x^{2}+2} d x $$

Short Answer

Expert verified
The evaluated integral of \(\int_{0}^{1} 5x e^{x^2 + 2} dx\) is \(\frac{5}{2}(e^3 - e^2)\).

Step by step solution

01

Identify a substitution

We can see that there is a composite function \(e^{x^2 + 2}\), and its internal function is \(x^2 + 2\). Let's substitute this internal function with a variable, say \(u\). So, let's set: \(u = x^2 + 2\)
02

Differentiate and isolate dx

Now, differentiate both sides with respect to the variable \(x\): \(\frac{du}{dx} = 2x\) Then, isolate the differential \(dx\) in terms of differential \(du\): \(dx = \frac{du}{2x}\)
03

Rewrite the integral using the substitution

Replace the function and differential \(dx\) in the integral with the substitution: \(\int_{0}^{1} 5x e^{u} (\frac{du}{2x})\) Now, simplify the integral: \(\int 5xe^{u}\cdot\frac{du}{2x} = \frac{5}{2}\int e^u du\)
04

Update the limits of integration

We need to change the limits of integration from \(x\)-values to \(u\)-values. Using our substitution \(u = x^2 + 2\), we have: For the lower limit, \(x = 0\): \(u = (0)^2 + 2 = 2\) For the upper limit, \(x = 1\): \(u = (1)^2 + 2 = 3\) So our new limits of integration are 2 and 3.
05

Evaluate the integral

Now we can integrate the simplified function: \(\frac{5}{2}\int_{2}^{3} e^u du\) The antiderivative of \(e^u\) is itself: \(\frac{5}{2}[e^u]_{2}^{3}\) Now, evaluate the antiderivative at the limits: \(\frac{5}{2}(e^3 - e^2)\)
06

Simplify the result

Finally, we have the numerical value of the integral: \(\int_{0}^{1} 5x e^{x^2 + 2} dx = \frac{5}{2}(e^3 - e^2)\) So, the evaluated integral is \(\frac{5}{2}(e^3 - e^2)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

U-Substitution
U-substitution is essentially the reverse process of the chain rule in differentiation and is a method used to evaluate integrals involving composite functions. It simplifies an integral by substiting a part of the integrand with a single variable, typically denoted as u.

Let's break down the steps involved using the example given:
  • Identifying the substitution: Spot the inner function within a composite function that would simplify the integral when replaced with u. In our case, the inner function is x^2 + 2.
  • Differentiating: You differentiate u with respect to x to find du/dx, which allows you to express dx in terms of du. Here, that gives us dx = du/(2x).
  • Rewriting the integral: Substitute u and dx into the integral. This also entails simplifying the integrand where possible, such as cancelling out common terms.
  • Adjusting the limits: If the integral is definite, we convert the x-limits to u-limits. In our example, x=0 becomes u=2 and x=1 turns into u=3.
Through these steps, we transformed a complicated integral into a simpler form that can be easily integrated.
Definite Integral
A definite integral represents the area under the curve of a function on a specific interval. In practical terms, it gives us the accumulated quantity, be it distance, area, or another measure based on the context, between two points along the curve of the function.

The process to solve a definite integral includes:
  • Finding the Antiderivative: Determine the general antiderivative or the indefinite integral of the function in question.
  • Evaluating at the boundaries: Replace the variable of the antiderivative with the upper and lower limits of the definite integral.
  • Calculating the difference: Subtract the value of the lower limit from the upper limit to find the total accumulated quantity.
For our example, after using U-Substitution, the definite integral became \(\frac{5}{2}\int_{2}^{3} e^u du\) with limits 2 and 3, making it straightforward to calculate the area under the curve of \(e^u\) from \(u=2\) to \(u=3\).
Antiderivative
An antiderivative, sometimes called an indefinite integral, is the inverse operation to taking the derivative and represents a family of functions whose derivative is the original function. Finding the antiderivative is central to solving both indefinite and definite integrals.

When we speak of evaluating integrals as seen in our example, we perform the following steps:
  • Integration: Determine the antiderivative of the function we are integrating. This step requires knowledge of integration rules and methods, including U-substitution.
  • Constant of Integration: For indefinite integrals, add a constant \(C\) because differentiating a constant gives zero, leaving the original function unchanged.
  • Applying the Fundamental Theorem of Calculus: For definite integrals, we apply the antiderivative to the bounds of the integral and subtract to find an exact value without the constant of integration.
The final step of our example's solution, where we evaluated \(\frac{5}{2}\left[e^u\right]_{2}^{3}\), involves applying the antiderivative \(e^u\) to the limits 2 and 3 to get the value \(\frac{5}{2}(e^3 - e^2)\).

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Most popular questions from this chapter

The normal distribution curve, which models the distributions of data in a wide range of applications, is given by the function $$ p(x)=\frac{1}{\sqrt{2 \pi} \sigma} e^{-(x-\mu)^{2} / 2 \sigma^{2}} $$ where \(\pi=3.14159265 \ldots\) and \(\sigma\) and \(\mu\) are constants called the standard deviation and the mean, respectively. Its graph (when \(\sigma=1\) and \(\mu=2\) ) is shown in the figure. Illustrate its use. In a survey, consumers were asked to rate a new toothpaste on a scale of \(1-10\). The resulting data are modeled by a normal distribution with \(\mu=4.5\) and \(\sigma=1.0\). The percentage of consumers who rated the toothpaste with a score between \(a\) and \(b\) on the test is given by $$ \int_{a}^{b} p(x) d x $$ a. Use a Riemann sum with \(n=10\) to estimate the percentage of customers who rated the toothpaste 5 or above. (Use the range \(4.5\) to \(10.5\).) b. What percentage of customers rated the toothpaste 0 or \(1 ?\) (Use the range \(-0.5\) to \(1.5\).)

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