Evaluate the integrals in Exercises 33–54.
51. ∫ from ln(π/6) to ln(π/2) 2e^v cos(e^v) dv

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Identify the integral to evaluate: \(\int_{\ln(\frac{\pi}{6})}^{\ln(\frac{\pi}{2})} 2e^{v} \cos(e^{v}) \, dv\).

Use the substitution method by letting \(u = e^{v}\). Then, compute the differential: \(du = e^{v} dv\), which implies \(dv = \frac{du}{u}\).

Rewrite the integral in terms of \(u\). Note that \(e^{v} = u\), so \(2e^{v} \cos(e^{v}) dv = 2u \cos(u) \cdot \frac{du}{u} = 2 \cos(u) du\).

Change the limits of integration according to the substitution: when \(v = \ln(\frac{\pi}{6})\), \(u = e^{\ln(\frac{\pi}{6})} = \frac{\pi}{6}\); when \(v = \ln(\frac{\pi}{2})\), \(u = e^{\ln(\frac{\pi}{2})} = \frac{\pi}{2}\).

Now, the integral becomes \(\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} 2 \cos(u) \, du\). Proceed to integrate \(2 \cos(u)\) with respect to \(u\).

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Key Concepts

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

Definite Integrals

A definite integral calculates the net area under a curve between two specific limits. It involves evaluating the integral function at the upper and lower bounds and subtracting these values. Understanding the limits of integration and their role in the evaluation process is essential.

Substitution Method

The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. It often involves identifying a function and its derivative within the integral, allowing for easier integration. This technique is crucial when dealing with composite functions like e^v inside trigonometric functions.

Integration of Composite Functions

Integrating composite functions requires recognizing inner and outer functions and applying appropriate techniques such as substitution. For example, integrating expressions like cos(e^v) involves treating e^v as the inner function and using substitution to simplify the integral.

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