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Chapter 6: Problem 40

Evaluate the integral. $$\int_{-2}^{0} x e^{-x^{2}} d x$$

Short Answer

Expert verified
The result of the definite integral is \(-1/2 [1 - e^4]\).

Step by step solution

01

Identify substitution

To make the integral easier to solve, we apply the technique of substitution. Let \(u = -x^2\). This means \(du = -2x dx\). We can then find \(dx\) in terms of \(du\) as \(dx = du / (-2x)\).
02

Substitute

Substituting \(u\) into the integral, and adjusting the limits of the integral accordingly, we get \(\int_{4}^{0} e^{u} du / -2\).
03

Simplify

This simplifies to \(-1/2 \int_{4}^{0} e^{u} du\).
04

Evaluate integral

The integral of \(e^{u}\) with respect to \(u\) is simply \(e^{u}\). So, we have \(-1/2 [e^{0} - e^{4}]\).
05

Simplify

Simplifying, we get \(-1/2 [1 - e^4]\).

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Key Concepts

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

Substitution Method
The substitution method is a powerful integration technique, frequently employed when dealing with complex integrals. The idea is to simplify an integral by substituting a part of it with a new variable, usually denoted as \( u \). This often reduces the complexity of the problem and makes the integration process more manageable.

To apply the substitution method, you should:
  • Identify a part of the integral that can be substituted, ideally something that will simplify the form of the function.
  • Define a new variable \( u \), and express \( dx \) in terms of \( du \) by differentiating \( u \) with respect to \( x \).
  • Adjust the limits of integration if you are dealing with definite integrals. Substitute these new limits in terms of \( u \).
  • Re-evaluate the integral in terms of \( u \) and solve it.
  • Finally, substitute back the original variable to arrive at the final result.
The substitution method can turn a complicated integral into a simpler one, as shown in our exercise where \( u = -x^2 \), simplifying the integral dramatically.
Definite Integrals
Definite integrals are a concept in calculus that allow us to calculate the net "area" under a curve from one point to another. Unlike indefinite integrals, which are more general, definite integrals have specified limits of integration, offering specific numerical results.

To solve a definite integral:
  • Evaluate the antiderivative of the function.
  • Apply the limits of integration to the antiderivative, which involves plugging in the upper limit and subtracting the value of plugging in the lower limit.
  • The result is the net area under the curve between those two points.
In the exercise, limits \([-2, 0]\) are transformed during substitution into \([4, 0]\), reflecting the change in variables from \( x \) to \( u \). This shift highlights how knowing how to adjust limits is crucial for correctly evaluating definite integrals.
Exponential Functions
Exponential functions, characterized by the form \( e^x \), are functions where the variable is in the exponent. These functions grow rapidly and have unique properties that simplify integration and differentiation.

Key aspects of exponential functions include:
  • The derivative of \( e^x \) is \( e^x \), and similarly, the integral of \( e^x \) is also \( e^x \).
  • They have a natural base \( e \), approximately equal to 2.718, which simplifies logarithmic and exponential operations.
  • In integration calculations, \( e^{u} \) makes solving certain integrals straightforward, as it is its own derivative and antiderivative.
During our exercise, using the substitution technique brought an exponential function \( e^u \) into focus. This highlights how exponential functions can make integral calculus problems easier when leveraged properly.

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Most popular questions from this chapter

Evaluate \(\int \frac{2}{x^{3}+x} d x\) in each of the following ways. First, use the substitution \(u=x^{2}+1\) and partial fractions. Second, use the substitution \(u=\frac{1}{x}\) and evaluate the resulting integral. Show that the two answers are equivalent.

Evaluate the integral. $$\int_{3}^{4} x \sqrt{x-3} d x$$

Find the median (another measure of average) for a random variable with pdf \(r e^{-r x}\) on \(x \geq 0 .\) The median is the \(50 \%\) mark on probability, that is, the value \(m\) for which $$ \int_{m}^{\infty} r e^{-r x} d x=\frac{1}{2} $$

As discussed in the chapter introduction, the mean (average) lifetime of a lightbulb might have the form \(\int_{0}^{\infty} 0.001 x e^{-0.001 x} d x .\) To determine the mean, compute \(\lim _{b \rightarrow \infty} \int_{0}^{b} 0.001 x e^{-0.001 x} d x\)

In exercises find the partial lractions decomposition. $$\frac{2 x^{3}-x^{2}}{\left(x^{2}+1\right)^{2}}$$
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