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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