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int (cos7x-cos8x)/(1+2cos5x)dx...

`int (cos7x-cos8x)/(1+2cos5x)dx`

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We have , `int(cos7x-cos8x)/(1+2cos5x)dx=1/2int(2sin(5x)/(2)cos7x-2sin(5x)/(2)cos8x)/(sin(5x)/(2)+2cos5xsin(5x)/(2))dx`
`1/2intsin(19x)/(2)-sin(9x)/(2)-(sin(21x)/(2)-sin(11x)/(2))/(sin(5x)/(2)+sin(15x)/(2)-sin(5x)/(2))dx`
`1/2int((sin(19x)/(2)+sin(11x)/(2)-(sin(21x)/(2)+sin(9x)/(2)))/(sin(15x)/(2)+sin(15x)/(2)-sin(5x)/(2)))dx`
`1/2int((sin(19x)/(2)+sin(11x)/(2))-(sin(21x)/(2)+sin(9x)/(2)))/(sin(15x)/(2))dx`
`1/2int(2sin(15x)/(2)cos2x-2sin(15x)/(2)cos3x)/(sin(15x)/(2))dx=int(cos2x-cos3x)dx=1/2sin2x-1/3sin3x+C`
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