## Problem Statement

Calculate the indefinite integral:

$$
\int \tan x \cdot \ln \cos x \, dx
$$
## Solution

$$
\int \operatorname{tg} x \cdot \ln \cos x d x=
$$

Substitution:

$$
\begin{aligned}
& y=\ln \cos x \\
& d y=\frac{1}{\cos x} \cdot(-\sin x) \cdot d x=-\operatorname{tg} x \cdot d x
\end{aligned}
$$

We get:

$$
=\int-y \cdot d y=-\frac{y^{2}}{2}+C=
$$

Reverse substitution:

$$
=-\frac{\ln ^{2} \cos x}{2}+C
$$

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+3-7$ >

Categories: Kuznetsov's Problem Book Integrals Problem 3 | Integrals

Ukrainian Banner Network

- Last modified on this page: 16:56, 1 March 2009.
- Content is available under CC-BY-SA 3.0.

Created by GeeTeatoo

## Problem Kuznetsov Integrals 3-8

## Material from Plusi