/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 22 Evaluate the integrals. $$ \... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 22

Evaluate the integrals. $$ \int_{-1}^{1} 3^{2 x-1} d x $$

Short Answer

Expert verified
The short version of the answer is: \(\int_{-1}^{1} 3^{2x-1} dx = \frac{53}{54}\).

Step by step solution

01

1. Identify the function

We are given the following integral to evaluate: $$ \int_{-1}^{1} 3^{2 x-1} d x $$ The function to be integrated is \(3^{2x-1}\).
02

2. Integrate the exponential function

To integrate \(3^{2x-1} dx\), we will make use of the integration rule for exponential functions: $$ \int e^{ax} dx = \frac{1}{a}e^{ax} + C $$ In our case, we'll replace \(e\) with \(3\) and \(a\) with \(2\). Also, since there's a "-1" in the exponent, we will factor it out: $$ \int 3^{2x} 3^{-1} dx = 3^{-1} \int 3^{2x} dx $$ Now apply the exponential integration rule: $$ 3^{-1} \int 3^{2x} dx = 3^{-1} \frac{1}{2} 3^{2x} + C = \frac{1}{6} 3^{2x} + C $$
03

3. Find the antiderivative with the given limits

Now that we have the antiderivative, let's apply the given limits of integration (-1 and 1). We'll substitute the limits into the antiderivative and find the difference: $$ \int_{-1}^{1} 3^{2x-1} dx = \left[ \frac{1}{6} 3^{2x} \right]_{-1}^{1} $$ Substitute the upper limit, 1: $$ \frac{1}{6} 3^{2(1)} = \frac{1}{6} 3^2 = \frac{9}{6} $$ Substitute the lower limit, -1: $$ \frac{1}{6} 3^{2(-1)} = \frac{1}{6} 3^{-2} = \frac{1}{6} \cdot \frac{1}{9} = \frac{1}{54} $$
04

4. Calculate the final result

Now, subtract the lower limit antiderivative from the upper limit antiderivative: $$ \int_{-1}^{1} 3^{2x-1} dx = \frac{9}{6} - \frac{1}{54} = \frac{27}{18} - \frac{1}{54} = \frac{53}{54} $$ So the integral evaluates to: $$ \int_{-1}^{1} 3^{2x-1} dx = \frac{53}{54} $$

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Key Concepts

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

Integration of Exponential Functions
When dealing with exponential functions, students often find the integration process slightly daunting. However, understanding the key pattern in these integrals can simplify the task considerably.

For exponential functions of the form \( e^{ax} \), where \( a \) is a constant, the integral can be easily found using the formula:

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Most popular questions from this chapter

The rate of change \(s(t)\) of the number of eBay stores can be approximated by \(s(t)=0.035 t^{4}-0.55 t^{3}-1.45 t^{2}+37 t-36\) stores/quarter \((1 \leq t \leq 14)\) \((t\) is time in quarters; \(t=1\) represents the start of the first quarter in 2005\()^{48}\) Use technology to compute \(\int_{5}^{13} s(t) d t\) correct to the nearest whole number. Interpret your answer.

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