Đáp án:
\[\int {f\left( x \right)dx} = \ln \left| {\sin x + \cos x} \right| + C\]
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
\int {f\left( x \right)} \\
= \int {\frac{{1 - 2{{\sin }^2}x}}{{2{{\sin }^2}\left( {x + \frac{\pi }{4}} \right)}}dx} \\
= \int {\frac{{{{\cos }^2}x + {{\sin }^2}x - 2{{\sin }^2}x}}{{2.{{\left( {\sin x.\cos \frac{\pi }{4} + \cos x.\sin \frac{\pi }{4}} \right)}^2}}}dx} \\
= \int {\frac{{{{\cos }^2}x - {{\sin }^2}x}}{{{{\left( {\sin x + \cos x} \right)}^2}}}dx} \\
= \int {\frac{{\left( {\cos x - \sin x} \right)\left( {\cos x + \sin x} \right)}}{{{{\left( {\sin x + \cos x} \right)}^2}}}dx} \\
= \int {\frac{{\cos x - \sin x}}{{\sin x + \cos x}}dx} \\
t = \sin x + \cos x \Rightarrow dt = \left( {\sin x + \cos x} \right)'dx = \left( {\cos x - \sin x} \right)dx\\
\Rightarrow \int {f\left( x \right)dx} = \int {\frac{{dt}}{t}} = \ln \left| t \right| + C = \ln \left| {\sin x + \cos x} \right| + C
\end{array}\)
