Evaluate the integrals in Exercises 31–78.
39. ∫(from 0 to π)tan(x/3)dx

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Identify the integral to be evaluated: \(\int_0^{\pi} \tan\left(\frac{x}{3}\right) \, dx\).

Use a substitution to simplify the integral. Let \(u = \frac{x}{3}\), which implies \(x = 3u\) and \(dx = 3 \, du\).

Change the limits of integration according to the substitution: when \(x = 0\), then \(u = 0\); when \(x = \pi\), then \(u = \frac{\pi}{3}\).

Rewrite the integral in terms of \(u\): \(\int_0^{\pi} \tan\left(\frac{x}{3}\right) \, dx = \int_0^{\frac{\pi}{3}} \tan(u) \cdot 3 \, du = 3 \int_0^{\frac{\pi}{3}} \tan(u) \, du\).

Recall the integral formula for tangent: \(\int \tan(u) \, du = -\ln|\cos(u)| + C\). Use this to express the integral and then apply the limits from \(0\) to \(\frac{\pi}{3}\).

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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 ∫ from a to b of f(x) dx, where a and b are the lower and upper bounds. Evaluating definite integrals often involves finding an antiderivative and then applying the Fundamental Theorem of Calculus.

Integration of Trigonometric Functions

Integrating trigonometric functions like tangent requires knowledge of their antiderivatives. For example, the integral of tan(x) is -ln|cos(x)| + C. Recognizing these standard forms helps simplify the integration process, especially when the argument of the function is a linear expression.

Substitution Method

The substitution method simplifies integrals by changing variables to transform the integral into a more familiar form. For integrals like ∫tan(x/3) dx, setting u = x/3 helps adjust the limits and integrand accordingly, making the integral easier to evaluate.

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