## Problem Statement

Calculate the definite integral:

$$
\int_{0}^{\arcsin (3 / \sqrt{10})} \frac{2 \tan x-5}{(4 \cos x-\sin x)^{2}} d x
$$
## Solution

$$
\begin{aligned}
& \arcsin (3 / \sqrt{10}) \quad \arcsin (3 / \sqrt{10}) \\
& \int_{0} \frac{2 \operatorname{tg} x-5}{(4 \cos x-\sin x)^{2}} d x=\quad \int_{0} \frac{2 \operatorname{tg} x-5}{16 \cos ^{2} x-8 \cos x \sin x+\sin ^{2} x} d x= \\
& \arcsin (3 / \sqrt{10}) \\
& =\int_{0} \frac{2 \operatorname{tg} x-5}{15 \cos ^{2} x-4 \sin 2 x+1} d x=
\end{aligned}
$$

We will use the substitution:

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

From which:

$$
\begin{aligned}
& \sin 2 x=\frac{2 t}{1+t^{2}}, \cos ^{2} x=\frac{1}{1+t^{2}}, d x=\frac{d t}{1+t^{2}} \\
& x=0 \Rightarrow t=\operatorname{tg} 0=0 \\
& x=\arcsin \frac{3}{\sqrt{10}} \Rightarrow t=\operatorname{tg}\left(\arcsin \frac{3}{\sqrt{10}}\right)=\sqrt{\frac{1}{1-\sin ^{2}\left(\arcsin \frac{3}{\sqrt{10}}\right)}-1}= \\
& =\sqrt{\frac{1}{1-\frac{9}{10}}-1}=\sqrt{10-1}=\sqrt{9}=3
\end{aligned}
$$

Substitute:

$$
\begin{aligned}
& =\int_{0}^{3} \frac{2 t-5}{\frac{15}{1+t^{2}}-\frac{8 t}{1+t^{2}}+1} \cdot \frac{d t}{1+t^{2}}=\int_{0}^{3} \frac{2 t-5}{15-8 t+1+t^{2}} d t= \\
& =\int_{0}^{3} \frac{2 t-8+3}{(t-4)^{2}} d t=2 \cdot \int_{0}^{3} \frac{1}{t-4} d t+3 \cdot \int_{0}^{3} \frac{1}{(t-4)^{2}} d t= \\
& =2 \ln \left|t-4 \|_{0}^{3}-\frac{3}{t-4}\right|_{0}^{3}=2 \ln |3-4|-2 \ln |0-4|-\frac{3}{3-4}+\frac{3}{0-4}=
\end{aligned}
$$

$$
=2 \ln 1-2 \ln 4+3-\frac{3}{4}=0-4 \ln 2+\frac{9}{4}=\frac{9}{4}-\ln 16
$$

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}+9-24 »

Categories: Kuznetsov's Problem Book Integrals Problem 9 | Integrals

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## Problem Kuznetsov Integrals $9-25$

## Material from PlusPi

This problem may have a different condition (possibly due to different editions or errors). For more details, see 9-25(2)