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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.3.39
Chapter 7, Problem 7.3.39

Evaluate the integrals in Exercises 33–54.
∫(from ln4 to ln9)e^(x/2)dx

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Identify the integral to be evaluated: \(\int_{\ln 4}^{\ln 9} e^{\frac{x}{2}} \, dx\).
Recognize that the integrand is an exponential function with a linear exponent. To integrate \(e^{\frac{x}{2}}\), use the substitution method or recall the integral formula for \(e^{ax}\), which is \(\frac{1}{a} e^{ax} + C\).
Set \(a = \frac{1}{2}\), so the antiderivative of \(e^{\frac{x}{2}}\) is \(2 e^{\frac{x}{2}} + C\).
Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper and lower limits: calculate \(2 e^{\frac{\ln 9}{2}} - 2 e^{\frac{\ln 4}{2}}\).
Simplify the expressions \(e^{\frac{\ln 9}{2}}\) and \(e^{\frac{\ln 4}{2}}\) by using the property \(e^{\ln a} = a\), rewriting them as \(9^{\frac{1}{2}}\) and \(4^{\frac{1}{2}}\) respectively, which correspond to the square roots of 9 and 4.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integral

A definite integral calculates the net area under a curve between two specific limits. It is represented as ∫ from a to b of f(x) dx, where a and b are the lower and upper bounds. Evaluating a definite integral involves finding the antiderivative and then applying the Fundamental Theorem of Calculus.
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Antiderivative of Exponential Functions

The antiderivative of an exponential function e^(kx) is (1/k)e^(kx) + C, where k is a constant. This rule allows us to reverse differentiation and find the original function before differentiation, which is essential for evaluating integrals involving exponentials.
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Properties of Natural Logarithms as Limits

Natural logarithms (ln) often appear as limits in integrals. Understanding that ln(a) is the power to which e must be raised to get a helps interpret the bounds. This knowledge is useful when substituting or simplifying expressions involving ln in definite integrals.
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