Evaluate the integrals in Exercises 47–68.

∫₀ ^π tan² (θ/3) dθ

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Recognize that the integral is \( \int_0^{\pi} \tan^2 \left( \frac{\theta}{3} \right) d\theta \). The integrand involves \( \tan^2(x) \), which can be rewritten using a trigonometric identity to simplify the integration.

Recall the identity: \( \tan^2 x = \sec^2 x - 1 \). Use this to rewrite the integral as \( \int_0^{\pi} \left( \sec^2 \left( \frac{\theta}{3} \right) - 1 \right) d\theta \).

Split the integral into two separate integrals: \( \int_0^{\pi} \sec^2 \left( \frac{\theta}{3} \right) d\theta - \int_0^{\pi} 1 \, d\theta \).

Make a substitution to handle the integral involving \( \sec^2 \left( \frac{\theta}{3} \right) \). Let \( u = \frac{\theta}{3} \), so that \( d\theta = 3 du \). Adjust the limits of integration accordingly: when \( \theta = 0 \), \( u = 0 \), and when \( \theta = \pi \), \( u = \frac{\pi}{3} \).

Rewrite the integral in terms of \( u \): \( 3 \int_0^{\frac{\pi}{3}} \sec^2 u \, du - \int_0^{\pi} 1 \, d\theta \). Then, integrate \( \sec^2 u \) which is \( \tan u \), and integrate the constant 1 over \( \theta \).

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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 specified limits. It is represented as ∫_a^b f(x) dx, where a and b are the limits of integration. Evaluating definite integrals often involves finding an antiderivative and then applying the Fundamental Theorem of Calculus.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values in their domains. For example, the identity tan²x = sec²x - 1 helps simplify integrals involving tan²x by rewriting them in terms of sec²x, which has a straightforward antiderivative.

Substitution Method

The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. For integrals like ∫ tan²(θ/3) dθ, substituting u = θ/3 helps adjust the limits and integrand, making the integral easier to evaluate.

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