Evaluate the integrals in Exercises 47–68.

∫⁰-π/3 sec x tan x dx

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Identify the integral to be evaluated: \(\int_{-\frac{\pi}{3}}^{0} \sec x \tan x \, dx\).

Recall the derivative of \(\sec x\) is \(\frac{d}{dx}(\sec x) = \sec x \tan x\), which matches the integrand.

Use this fact to rewrite the integral as \(\int_{-\frac{\pi}{3}}^{0} \frac{d}{dx}(\sec x) \, dx\).

Apply the Fundamental Theorem of Calculus, which states that \(\int_a^b f'(x) \, dx = f(b) - f(a)\), so the integral becomes \(\sec x \big|_{-\frac{\pi}{3}}^{0}\).

Evaluate \(\sec x\) at the upper and lower limits: calculate \(\sec(0)\) and \(\sec\left(-\frac{\pi}{3}\right)\), then subtract accordingly.

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