Evaluate the integrals in Exercises 47–68.

∫₀^π/3 sec² θ dθ

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Identify the integral to be evaluated: \(\int_0^{\frac{\pi}{3}} \sec^2 \theta \, d\theta\).

Recall the antiderivative of \(\sec^2 \theta\) is \(\tan \theta\), since \(\frac{d}{d\theta}(\tan \theta) = \sec^2 \theta\).

Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper and lower limits: \(\tan \theta \Big|_0^{\frac{\pi}{3}}\).

Substitute the limits into the antiderivative expression: calculate \(\tan \left( \frac{\pi}{3} \right) - \tan(0)\).

Simplify the expression using known values of tangent at these angles to find the value of the definite integral.

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Key Concepts

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

Definite Integral

A definite integral calculates the net area under a curve between two specified limits. It is represented as ∫_a^b 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 sec² θ

The antiderivative of sec² θ is tan θ, since the derivative of tan θ is sec² θ. Recognizing this allows direct integration of sec² θ, simplifying the evaluation of the integral without complex substitutions.

Fundamental Theorem of Calculus

This theorem connects differentiation and integration, stating that if F is an antiderivative of f on [a, b], then ∫_a^b f(x) dx = F(b) - F(a). It provides a method to evaluate definite integrals by computing the difference of antiderivative values at the bounds.

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