Tìm nguyên hàm của hàm số sau: ∫ln(cosx)cos2xdx

Question

\(\int \frac{\ln \left(\cos {\it x}\right)}{\cos ^{2} {\it x}} d{\it x} \)

Answer 1

Đặt \(\left\{\begin{array}{l} {{\it u}=\ln \left(\cos {\it x}\right)} \\ {d{\it v}=\frac{1}{\cos ^{2} {\it x}} d{\it x}} \end{array}\right. \Rightarrow \left\{\begin{array}{l} {d{\it u}=\frac{\left(\cos {\it x}\right)^{{'} } }{\cos {\it x}} d{\it x}=-\tan {\it x}.d{\it x}} \\ {v=\tan {\it x}} \end{array}\right. \).
\(\begin{array}{l} {\Rightarrow \int \frac{\ln \left(\cos {\it x}\right)}{\cos ^{2} {\it x}} d{\it x} =\tan {\it x}.\ln \left(\cos {\it x}\right)+\int \tan ^{2} {\it x}.d{\it x} } \\ {\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, =\tan {\it x}.\ln \left(\cos {\it x}\right)+\int \left(\tan ^{2} {\it x}+1-1\right)d{\it x} } \\ {\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, =\tan {\it x}.\ln \left(\cos {\it x}\right)+\tan {\it x}-{\it x}+C} \end{array}\)