## Problem Statement

Calculate the definite integral:

$$
\int_{-\arcsin (2 / \sqrt{5})}^{\pi / 4} \frac{2-\operatorname{tg} x}{(\sin x+3 \cos x)^{2}} d x
$$
## Solution

$$
\begin{aligned}
& \int_{-\arcsin (2 / \sqrt{5})}^{\pi / 4} \frac{2-\operatorname{tg} x}{(\sin x+3 \cos x)^{2}} d x=\int_{-\arcsin (2 / \sqrt{5})}^{\pi / 4} \frac{2-\operatorname{tg} x}{\sin ^{2} x+6 \sin x \cos x+9 \cos ^{2} x} d x= \\
& =\int_{\arcsin (\rho / \sqrt{5})}^{\pi / 4} \frac{2-\operatorname{tg} x}{1+3 \sin 2 x+8 \cos ^{2} x} d x=
\end{aligned}
$$

We will use the substitution:

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

From which:

$$
\begin{aligned}
& \sin 2 x=\frac{2 t}{1+t^{2}}, \cos ^{2} x=\frac{1}{1+t^{2}}, d x=\frac{d t}{1+t^{2}} \\
& x=-\arcsin \frac{2}{\sqrt{5}} \Rightarrow t=\operatorname{tg}\left(-\arcsin \frac{2}{\sqrt{5}}\right)=\operatorname{tg}\left(-\arcsin \frac{2}{\sqrt{5}}\right)= \\
& =-\sqrt{\frac{1}{1-\sin ^{2}\left(\arcsin \frac{2}{\sqrt{5}}\right)}-1}=-\sqrt{\frac{1}{1-\frac{4}{5}}-1}=-\sqrt{5-1}=-\sqrt{4}=-2 \\
& x=\frac{\pi}{4} \Rightarrow t=\operatorname{tg} \frac{\pi}{4}=1
\end{aligned}
$$

Substitute:

$$
\begin{aligned}
& =\int_{-2}^{1} \frac{2-t}{1+\frac{6 t}{1+t^{2}}+\frac{8}{1+t^{2}}} \cdot \frac{d t}{1+t^{2}}=\int_{-2}^{1} \frac{2-t}{1+t^{2}+6 t+8} d t=\int_{-2}^{1} \frac{2-t}{(t+3)^{2}} d t= \\
& =-\int_{-2}^{1} \frac{t+3-5}{(t+3)^{2}} d t=-\int_{-2}^{1} \frac{1}{t+3} d t+\int_{-2}^{1} \frac{5}{(t+3)^{2}} d t= \\
& =-\ln \left|t+3 \|_{-2}^{1}-\frac{5}{t+3}\right|_{-2}^{1}=
\end{aligned}
$$

$$
\begin{aligned}
& =-\ln |1+3|+\ln |-2+3|-\frac{5}{1+3}+\frac{5}{-2+3}= \\
& =-\ln 4+\ln 1-\frac{5}{4}+5=\frac{15}{4}-\ln 4
\end{aligned}
$$

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\%D1\%82 $\% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}$ _9-27"

Categories: Kuznetsov's Integral Problems 9 | Integrals

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## Problem Kuznetsov Integrals 9-28

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