Answer to Question #227619 in Calculus for Moe

Question

Which of the following values represent the definite ${\int_{\pi/2}^{2\pi/3}\frac{1}{1+\cos{\theta}}d\theta}$

1) e

2) 1

3) $\sqrt{3}$

4) $\sqrt{3}-1$

Answer 1

$$\displaystyle\int_{\pi/2}^{2\pi/3}\dfrac{1}{1+\cos \theta}d\theta=\displaystyle\int_{\pi/2}^{2\pi/3}\dfrac{1}{2\cos^2(\dfrac{\theta}{2})}d\theta$$

$$=[\tan(\dfrac{\theta}{2})]\begin{matrix} 2\pi/3 \\ \pi/2 \end{matrix}=\tan(\pi/3)-\tan(\pi/4)=\sqrt{3}-1$$

4) $\sqrt{3}-1$