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

Verified step by step guidance

1. Identify the integral to be evaluated: \(\int_{\ln 4}^{\ln 9} e^{\frac{x}{2}} \, dx\).
2. 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 \(\frac{1}{a} e^{ax} + C\).
3. Set \(a = \frac{1}{2}\), so the antiderivative of \(e^{\frac{x}{2}}\) is \(2 e^{\frac{x}{2}} + C\).
4. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper and lower limits: calculate \(2 e^{\frac{\ln 9}{2}} - 2 e^{\frac{\ln 4}{2}}\).
5. 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.

Key Concepts

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.

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.

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.