Question

Help please!!

integrate the following:

`\displaystyle \int ( \cos ^(2) (x) - \cos ^(2) ( \alpha ) )/( \cos( x) - \cos( \alpha ) )`
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Answer 1

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

Explanation:

we are given

`\displaystyle \int ( \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 ( ((\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`