Question

Evaluate the integrals in Exercises 39–56.43. ∫(from 0 to π)(sin t)/(2 - cos t) dt

Accepted Answer

Recognize that the integral is a definite integral of the form $$\$$int_0$$^{\pi} \frac{\sin t}{2 - \cos t} \, dt. $$Our goal is to find a way to simplify or transform the integrand to make it easier to integrate. Consider using a substitution to simplify the integral. Since the integrand involves $$\sin t$$ and $$\cos t$$, a natural substitution is to let $$u = 2 - \cos t. $$Then, compute $$du in $$terms of $$dt. $$Calculate the derivative: $$\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. $$Change the limits of integration accordingly: when $$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$$^3 \frac{1}{u} \, du. $$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.

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

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주요 개념

Definite Integrals

Integration Techniques for Trigonometric Functions

Substitution Method