## Problem Statement

Calculate the definite integral:

$$
\int_{0}^{\frac{2 \pi}{3}} \frac{1+\sin x}{1+\cos x+\sin x} d x
$$
## Solution

We will use the universal substitution:

$$
t=\operatorname{tg} \frac{x}{2}
$$

From which:

$$
\begin{aligned}
& \sin x=\frac{2 t}{1+t^{2}}, \cos x=\frac{1-t^{2}}{1+t^{2}}, d x=\frac{2 d t}{1+t^{2}} \\
& x=0 \Rightarrow t=\operatorname{tg} \frac{0}{2}=\operatorname{tg} 0=0 \\
& x=\frac{2 \pi}{3} \Rightarrow t=\operatorname{tg} \frac{\left(\frac{2 \pi}{3}\right)}{2}=\operatorname{tg} \frac{\pi}{3}=\sqrt{3}
\end{aligned}
$$

Substitute:

$$
\begin{aligned}
& \int_{0}^{\frac{2 \pi}{3}} \frac{1+\sin x}{1+\cos x+\sin x} d x=\int_{0}^{\sqrt{3}} \frac{\left(1+\frac{2 t}{1+t^{2}}\right) \cdot \frac{2 d t}{1+t^{2}}}{1+\frac{1-t^{2}}{1+t^{2}}+\frac{2 t}{1+t^{2}}}=\int_{0}^{\sqrt{3}} \frac{\frac{1+t^{2}+2 t}{1+t^{2}} \cdot \frac{2 d t}{1+t^{2}}}{\frac{1+t^{2}+1-t^{2}+2 t}{1+t^{2}}}= \\
& =\int_{0}^{\sqrt{3}} \frac{(1+t)^{2} \cdot \frac{2 d t}{1+t^{2}}}{2(1+t)}=\int_{0}^{\sqrt{3}} \frac{(1+t) d t}{1+t^{2}}= \\
& =\int_{0}^{\sqrt{3}}\left(\frac{d t}{1+t^{2}}+\frac{t \cdot d t}{1+t^{2}}\right)=\int_{0}^{\sqrt{3}} \frac{d t}{1+t^{2}}+\frac{1}{2} \cdot \int_{0}^{\sqrt{3}} \frac{d\left(t^{2}+1\right)}{1+t^{2}}= \\
& =\left.\operatorname{arctg}\right|_{0} ^{\sqrt{3}}+\left.\frac{1}{2} \cdot \ln \left|t^{2}+1\right|\right|_{0} ^{\sqrt{3}}= \\
& =\operatorname{arctg}(\sqrt{3})-\operatorname{arctg} 0+\frac{1}{2} \cdot \ln \left|(\sqrt{3})^{2}+1\right|-\frac{1}{2} \cdot \ln \left|0^{2}+1\right|= \\
& =\frac{\pi}{3}-0+\frac{1}{2} \cdot \ln 4-\frac{1}{2} \cdot \ln 1=\frac{\pi}{3}+\ln 2
\end{aligned}
$$

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+8-10$ »

Categories: Kuznetsov's Problem Book Integrals Problem 8 | Integrals

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## Problem Kuznetsov Integrals 8-11

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