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Chapter 6: Problem 5

Evaluate the integral. $$\int e^{3-2 x} d x$$

Short Answer

Expert verified
The integral of \( e^{3-2x} \) is \( -\frac{1}{2}e^{3-2x} + C \)

Step by step solution

01

Identify the integral

The given function to integrate is \( e^{3-2x} \), which is of the form \( e^{ax+b} \). Identify a = -2 and b = 3.
02

Apply the integral formula

The next step is to apply the formula of integral of \( e^{ax+b} \), which is \( \frac{1}{a}e^{ax+b} \). Substituting a = -2, we get \( -\frac{1}{2}e^{3-2x} \).
03

Evaluate the integral

Finally, applying the formula, we have \( \int e^{3-2x} dx = -\frac{1}{2}e^{3-2x} + C \), where C is the constant of integration.

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Key Concepts

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

Indefinite Integral
When dealing with calculus, the concept of an indefinite integral is fundamental. It refers to the process of finding a function whose derivative equals the given function. This process is known as anti-differentiation or integration. For the given exercise,
Exponential Integration Techniques
Exponential functions, represented as e^x, where 'e' is the base of natural logarithms, appear frequently in various fields such as physics, engineering, and mathematics.
Integration by Substitution
Integration by substitution is a technique used to evaluate integrals that may not initially be in a form that is easy to integrate.

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Most popular questions from this chapter

In exercises find the partial fractions decomposition and an antiderivative. If you have a CAS available, use it to check your answer. $$\frac{x^{3}+x}{3 x^{2}+2 x+1}$$

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