Answer:
[tex] \displaystyle \sin(x) + x\cos( \alpha ) + \rm C[/tex]
Step-by-step explanation:
we are given
[tex] \displaystyle \int \frac{ \cos ^{2} (x) - \cos ^{2} ( \alpha ) }{ \cos( x) - \cos( \alpha ) } dx[/tex]
we want to Integrate it
notice that, the denominator can be rewritten by using algebraic identity
remember that,
[tex] \displaystyle {a}^{2} - {b}^{2} = ( a + b)(a - b)[/tex]
therefore
rewrite the denominator by using the identity:
[tex] \displaystyle \int \frac{ ((\cos (x) + \cos ( \alpha) ) ( \cos(x) - \cos( \alpha ) ) }{ \cos( x) - \cos( \alpha ) }dx [/tex]
reduce fraction:
[tex] \displaystyle \int \frac{ ((\cos (x) + \cos ( \alpha) ) ( \cancel{ \cos(x) -\cos( \alpha ) ) }}{ \cancel{\cos( x) - \cos( \alpha ) } }dx[/tex]
[tex] \displaystyle \int \cos(x) + \cos( \alpha ) dx[/tex]
recall that,
[tex] \rm\displaystyle \int f(x) \pm g(x)dx =\int f(x) \pm \int g(x)dx[/tex]
by that
we get
[tex] \displaystyle \int \cos(x) dx + \int \cos( \alpha )dx [/tex]
we also know that,
[tex] \displaystyle \int \cos(x) = \sin(x)dx [/tex]
and also the Integration of a constant is always cx
so,
[tex] \displaystyle \sin(x) + x\cos( \alpha ) [/tex]
and of course we have to add constant
[tex] \displaystyle \sin(x) + x\cos( \alpha ) + \rm C[/tex]
