## Problem Statement

Calculate the area of the figure bounded by the graphs of the functions:

$$
y=\frac{1}{1+\cos x}, y=0, x=\frac{\pi}{2}, x=-\frac{\pi}{2}
$$
## Solution

$$
\begin{aligned}
S= & \int_{-\pi / 2}^{\pi / 2} \frac{1}{1+\cos x} d x= \\
& =\int_{-\pi / 2}^{\pi / 2} \frac{1}{2 \cos ^{2} \frac{x}{2}} d x= \\
& =\int_{-\pi / 2}^{\pi / 2} \frac{1}{\cos ^{2} \frac{x}{2}} d\left(\frac{x}{2}\right)= \\
& =\left.\operatorname{tg}_{\frac{1}{2}}\right|_{-\pi / 2} ^{\pi / 2}=1-(-1)=2
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_aa6951b1e92815ea34b6g-38.jpg?height=934&width=1154&top_left_y=1098&top_left_x=751)

Source — «http://pluspi.org/wiki/index.php/\�\�\�\�\�\�\�\�\�\�\�\� \%D0\%9A\%D1\%83\%D0\%B7\%D0\%BD\%D0\%B5\%D1\%86\%D0\%BE\%D0\%B2_\%D0\%98\%D0\%BD $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+14-20$ » Categories: Kuznetsov Problem Book Integrals Problem 14 | Integrals

- Last edited: 08:08, June 11, 2010.
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Created by GeeTeatoo

## Problem Kuznetsov Integrals 14-21

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