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Chapter 9: Problem 22

\(\int e^{3 x} e^{2 x} d x\)

Short Answer

Expert verified
\( \int e^{5x} \, dx = \frac{1}{5}e^{5x} + C \)

Step by step solution

01

Combine the Exponents

Use the exponent rule for multiplication, which states that when multiplying like bases, you add the exponents. Combine the exponents \(3x\) and \(2x\) in the integrand: \[ e^{3x} \times e^{2x} = e^{(3x + 2x)} = e^{5x} \]
02

Set Up the Integral

Rewrite the integral with the simplified exponent: \[ \text{Now, the integral becomes } \ \ \ \ \int e^{5x} \, dx \]
03

Integrate the Exponential Function

Use the integral formula for an exponential function \(\int e^{ax} \, dx = \frac{1}{a}e^{ax} + C\). Here, \(a = 5\): \[ \text{So, } \ \ \ \ \int e^{5x} \, dx = \frac{1}{5}e^{5x} + C \]

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Key Concepts

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

Exponential Functions
Exponential functions are crucial in calculus and many other math disciplines. They are represented by an equation of the form \(e^{kx}\), where \(e\) is the base and \(kx\) is the exponent. The base \(e\) is an irrational number, approximately equal to 2.71828. Exponential functions are widely used because they model growth and decay processes efficiently.
In this exercise, the integral involves the product of two exponential functions, \(e^{3x}\) and \(e^{2x}\), which need to be simplified before integration.
By learning exponential functions, students can understand complex real-world phenomena, like population growth, radioactive decay, and interest calculations.
Integration Techniques
Integration techniques are methods used to solve integrals, which are a fundamental concept in calculus. Integrals allow us to find areas under curves and accumulate quantities. There are multiple techniques, and the choice depends on the form of the function. In this particular exercise, we use the straightforward technique of integrating exponential functions.
The strategy involves:
  • Simplifying the integrand using exponent rules, if necessary.
  • Applying known integral formulas, such as \( \int e^{ax} \,dx = \frac{1}{a} e^{ax} + C \).

Properly using these techniques makes solving integrals simpler and more intuitive.
Mastering a variety of integration methods, such as substitution, partial fractions, and integration by parts, is essential for tackling more complex problems in integral calculus.
Exponent Rules
Understanding exponent rules is crucial when dealing with exponential functions. These rules simplify expressions and are immensely helpful in calculus. The main rules to remember are:
  • Product of like bases: When multiplying like bases, add the exponents.
    \ e^{a} \cdot e^{b} = e^{a + b} \
  • Quotient of like bases: When dividing like bases, subtract the exponents.
    \ e^{a} \/ e^{b} = e^{a - b} \
  • Power of a power: When raising a power to another power, multiply the exponents.
    \ (e^a)^b = e^{a\cdot b} \

In the given exercise, combining the exponents of \( e^{3x} \) and \( e^{2x} \) using the product rule simplifies the integral into a form that is easier to integrate:
\ e^{3x} \cdot e^{2x} = e^{3x + 2x} = e^{5x} \

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Most popular questions from this chapter

Find the area of the region bounded by the curve \(y=2 /(x-3)\), the \(x\) axis, and the lines \(x=4\) and \(x=5\).

When a simple electric circuit, containing no condensers but having inductance and resistance, has the electromotive force removed, the rate of decrease of the current is proportional to the current. The current is \(i\) amperes \(t\) sec after the cutoff, and \(i=40\) when \(t=0\). If the current dies down to 15 amperes in \(0.01 \mathrm{sec}\), find \(i\) in terms of \(t\).

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In a telegraph cable, the measure of the speed of the signal is proportional to \(x^{2} \ln (1 / x)\), where \(x\) is the ratio of the measure of the radius of the core of the cable to the measure of the thickness of the cable's winding. Find the value of \(x\) for which the speed of the signal is greatest.

\(\int \frac{5-4 x^{2}}{3+2 x} d x\)
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