Evaluate the integrals in Exercises 39–56.
43. ∫(from 0 to π)(sin t)/(2 - cos t) dt

Verified step by step guidance

1. Recognize that the integral is a definite integral of the form \(\int_0^{\pi} \frac{\sin t}{2 - \cos t} \, dt\). Our goal is to find a way to simplify or transform the integrand to make it easier to integrate.
2. Consider using a substitution to simplify the integral. Since the integrand involves \(\sin t\) and \(\cos t\), a natural substitution is to let \(u = 2 - \cos t\). Then, compute \(du\) in terms of \(dt\).
3. Calculate the derivative: \(\frac{du}{dt} = \sin t\), so \(du = \sin t \, dt\). This means that \(\sin t \, dt = du\), which allows us to rewrite the integral in terms of \(u\).
4. Change the limits of integration accordingly: when \(t = 0\), \(u = 2 - \cos 0 = 2 - 1 = 1\); when \(t = \pi\), \(u = 2 - \cos \pi = 2 - (-1) = 3\). Substitute these into the integral to get \(\int_1^3 \frac{1}{u} \, du\).
5. Integrate \(\int_1^3 \frac{1}{u} \, du\), which is a standard integral resulting in \(\ln|u|\) evaluated from 1 to 3. After integration, substitute back the limits to express the answer in terms of natural logarithms.

Key Concepts

Definite Integrals

A definite integral calculates the net area under a curve between two specified limits. It is evaluated by finding an antiderivative and then applying the Fundamental Theorem of Calculus to compute the difference at the upper and lower bounds.

Integration Techniques for Trigonometric Functions

Integrating expressions involving sine and cosine often requires techniques such as substitution, trigonometric identities, or rewriting the integrand to simplify the integral. Recognizing patterns and applying identities like sin²x + cos²x = 1 can be crucial.

Substitution Method

The substitution method involves changing variables to simplify an integral. By letting a new variable equal a function inside the integral, the integral can be transformed into a more manageable form, making it easier to integrate.