Question

In Exercises \(27-40\) , evaluate each integral using Part 2 of the Fundamental Theorem. Support your answer with NINT if you are unsure. $$\int_{\pi / 6}^{5 \pi / 6} \csc ^{2} \theta d \theta$$

Short answer

The result of \(\int_{\pi / 6}^{5 \pi / 6} \csc ^{2} \theta d \theta\) is \(1/\sqrt{3} + \sqrt{3}\)

Step-by-step solution

Find the antiderivative of the function

The antiderivative of \( \csc^{2} \theta \) is \( -\cot \theta \) . This can be memorized as a standard rule, or derived by knowing that \( \csc^{2} \theta = 1 + \cot^{2} \theta \) and using the chain rule to find the derivative of \( \cot \theta \) .

Evaluate the antiderivative at the upper and lower bounds of the integral

We now evaluate \( -\cot \theta \) at \( 5\pi/6 \) and \( \pi/6 \). We get \( -\cot(5\pi/6) = 1/\sqrt{3} \) and \( -\cot(\pi/6) = -\sqrt{3} \) .

Apply Part 2 of the Fundamental Theorem of Calculus

According to the Fundamental Theorem, we subtract the lower bound result from the upper bound result: \( 1/\sqrt{3} - (-\sqrt{3}) = 1/\sqrt{3} + \sqrt{3} \) .