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Chapter 7: Problem 5

Find the integrals. Check your answers by differentiation. $$\int e^{-x} d x$$

Short Answer

Expert verified
The integral is \( -e^{-x} + C \), where \( C \) is a constant.

Step by step solution

01

Identify the Form of the Integral

The integral \( \int e^{-x} \, dx \) is a straightforward exponential integral of the form \( \int e^{ax} \, dx \), where the solution is \( \frac{1}{a} e^{ax} + C \). In this case, \( a = -1 \).
02

Solve the Integral

Using the formula for exponential integrals, substitute \( a = -1 \). The integral becomes \( \int e^{-x} \, dx = \frac{1}{-1} e^{-x} + C = -e^{-x} + C \), where \( C \) is the constant of integration.
03

Differentiate to Check the Solution

Differentiate the function \( -e^{-x} + C \) to verify the solution. The derivative of \( -e^{-x} \) is \( e^{-x} \), because the derivative of \( e^{-x} \) is \( -e^{-x} \) and the negative sign switches it back to \( e^{-x} \). The derivative of a constant \( C \) is zero. Hence, the derivative of \( -e^{-x} + C \) is \( e^{-x} \).
04

Confirm the Solution Matches the Original Integrand

Since the derivative \( e^{-x} \) matches the original integrand, \( e^{-x} \), the solution to the integral is confirmed to be correct.

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Key Concepts

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

Exponential Integral
Exponential integrals are a common mathematical function used in calculus involving integrals of exponential functions. The expression used in the exercise, \( \int e^{-x} \, dx \), is a classic example of an exponential integral. When integrating an exponential function of the form \( \int e^{ax} \, dx \), the antiderivative will be \( \frac{1}{a} e^{ax} + C \). Here, \( a \) is a constant that appears as a coefficient in the exponent.
  • In our integral \( e^{-x} \), the coefficient \( a \) is \(-1\).
  • Applying the formula, we find that the integral evaluates to \( -e^{-x} + C \).
  • The function \( e^{-x} \) rapidly decreases as \( x \) increases, a characteristic feature of negative-exponent exponential functions.
Understanding exponential integrals is essential when dealing with real-world problems involving exponential growth and decay, such as in population models or radioactive decay.
Differentiation
Differentiation is the mathematical process used to find the derivative of a function. In calculus, differentiating an integral is a common way to verify its correctness. The derivative provides information about the rate of change of the function.
  • To check an integration solution, you differentiate the result and see if you return to the original function inside the integral.
  • In the solved exercise, differentiating \( -e^{-x} + C \) yields \( e^{-x} \), precisely the original integrand.
  • The process ensures we've correctly found the antiderivative by confirming that when differentiated, it returns to the beginning function.
Differentiation is a powerful tool not only for checking integrals but also for analyzing and understanding functions.
Constant of Integration
The constant of integration, represented as \( C \), frequently appears in indefinite integrals. It is a placeholder that represents any constant value.
  • After integrating, the general solution includes \( C \) because indefinite integrals represent families of functions.
  • These families differ by a constant, since differentiating a constant results in zero.
  • Thus, regardless of the value of \( C \), the derivative of the integral \(-e^{-x} + C\) will remain \( e^{-x} \).
The constant of integration is vital for capturing all potential solutions to an integral. In physical and applied sciences, initial conditions or boundary values often determine the specific value of \( C \). In the context of pure mathematics, specifying \( C \) acknowledges the integral's general nature.

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Most popular questions from this chapter

Calculate the integrals in Exercises \(5-33,\) if they converge. You may calculate the limits by appealing to the dominance of one function over another, or by l'Hopital's rule. $$\int_{0}^{\pi} \frac{1}{\sqrt{x}} e^{-\sqrt{x}} d x$$

Give an example of: An indefinite integral involving sin \(x\) that can be evaluated with a reduction formula.

For Problems \(89-90,\) which technique is useful in evaluating the integral? (a) Integration by parts (b) Partial fractions (c) Long division (d) Completing the square (e) \(\quad\) A trig substitution (f) Other substitutions $$\int \frac{x^{2}}{\sqrt{1-x^{2}}} d x$$

Decide whether the statements are true or false. Give an explanation for your answer. When integrating by parts, it does not matter which factor we choose for \(u.\)

The Law of Mass Action tells us that the time, \(T\), taken by a chemical to create a quantity \(x_{0}\) of the product (in molecules) is given by $$T=\int_{0}^{x_{0}} \frac{k d x}{(a-x)(b-x)}$$ where \(a\) and \(b\) are initial quantities of the two ingredients used to make the product, and \(k\) is a positive constant. Suppose \(0
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