Question

Help please!!

Answer 1

Answer:

$\displaystyle \sin(x) + x\cos( \alpha ) + \rm C$

Step-by-step explanation:

we are given

$\displaystyle \int \frac{ \cos ^{2} (x) - \cos ^{2} ( \alpha ) }{ \cos( x) - \cos( \alpha ) } dx$

we want to Integrate it

notice that, the denominator can be rewritten by using algebraic identity

remember that,

$\displaystyle {a}^{2} - {b}^{2} = ( a + b)(a - b)$

therefore

rewrite the denominator by using the identity:

$\displaystyle \int \frac{ ((\cos (x) + \cos ( \alpha) ) ( \cos(x) - \cos( \alpha ) ) }{ \cos( x) - \cos( \alpha ) }dx$

reduce fraction:

$\displaystyle \int \frac{ ((\cos (x) + \cos ( \alpha) ) ( \cancel{ \cos(x) -\cos( \alpha ) ) }}{ \cancel{\cos( x) - \cos( \alpha ) } }dx$

$\displaystyle \int \cos(x) + \cos( \alpha ) dx$

recall that,

$\rm\displaystyle \int f(x) \pm g(x)dx =\int f(x) \pm \int g(x)dx$

by that

we get

$\displaystyle \int \cos(x) dx + \int \cos( \alpha )dx$

we also know that,

$\displaystyle \int \cos(x) = \sin(x)dx$

and also the Integration of a constant is always cx

so,

$\displaystyle \sin(x) + x\cos( \alpha )$

and of course we have to add constant

$\displaystyle \sin(x) + x\cos( \alpha ) + \rm C$