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Question
Integrate the following with respect to the respective variable : `(cos 7x - cos8x)/(1 + 2 cos 5x)`
Solution
`int (cos 7x - cos8x)/(1 + 2 cos 5x)*dx`
= `int (sin5x(cos7x - cos8x))/(sin5x(1 + 2 cos5x))*dx`
= `int (sin5x (cos7x - cos8x))/(sin5x + 2 sin 5x cos5x)*dx`
= `int (sin5x(cos7x - cos8x))/(sin5x + sin 10x)*dx`
= `int (2sin(5x/2)*cos ((5x)/2) xx 2sin ((7x + 8x)/2)*sin((8x - 7x)/2))/(2sin ((10x + 5x)/2)*cos ((10x - 5x)/2))*dx`
= `int (2sin ((5x)/2)*cos((5x)/2) xx 2sin((15x)/2)*sin(x/2))/(2sin((15x)/2)*cos((5x)/2))*dx`
= `int 2sin ((5x)/2)*sin(x/2)*dx`
= `int[cos ((5x)/2 - x/2) - cos((5x)/2 + x/2)]*dx`
= `int (cos 2x - cos 3x)*dx`
= `int cos2x*dx - int cos3*dx`
= `(sin2x)/(2) - (sin3x)/(3) + c`.
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