Question

The solution to this question please

Answer 1

see the numerator ,write it as
$\cos(x) = {cos}^{2} ( \frac{x}{2} ) - {sin}^{2} ( \frac{x}{2} ) \\ \: \: \: \: = ( \cos( \frac{x}{2} ) + \sin( \frac{x}{2} ) )( \cos( \frac{x}{2} ) - \sin( \frac{x}{2} ) )$
your question becomes integration of
$\frac{ \cos( \frac{x}{2} ) - \sin( \frac{x}{2} ) }{( \cos(( \frac{x}{2} )) + \sin( \frac{x}{2} ) ) {}^{2} } dx \\ let \: \cos( \frac{x}{2} ) + \sin( \frac{x}{2} ) = t \\( - ( \frac{1}{2} ) \sin( \frac{x}{2} ) + ( \frac{1}{2} ) \cos( \frac{x}{2} ) )dx = dt \\ ( \cos( \frac{x}{2} ) - \sin( \frac{x}{2} ) ) dx= 2dt$
your question Now becomes integration of

$\frac{2dt}{ {t}^{2} } \\ = 2\frac{t {}^{ - 2 + 1} }{ - 2 + 1} \\ = - \frac{2}{t} + c \\ = \frac{ - 2}{ \cos( \frac{x}{2} ) + \sin( \frac{x}{2} ) } + c$
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