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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Key Concepts

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

Integration of Trigonometric Functions

This involves finding the antiderivative of functions involving trigonometric expressions like sec x and tan x. Recognizing standard integral forms helps simplify the process, such as knowing that the derivative of sec x is sec x tan x.

Definite Integrals and Limits of Integration

Definite integrals calculate the net area under a curve between two specific points, called limits of integration. Understanding how to apply these limits after finding the antiderivative is essential to evaluate the integral's exact value.

Fundamental Theorem of Calculus

This theorem connects differentiation and integration, stating that if F is an antiderivative of f, then the definite integral of f from a to b equals F(b) - F(a). It provides the method to evaluate definite integrals using antiderivatives.

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