Evaluate: ∫sin(x-a)sin(x+a)dx - Mathematics and Statistics

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Sum

Evaluate:

`int (sin(x - a))/(sin(x + a))dx`

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Solution

Let I = `int (sin(x - a))/(sin(x + a))dx`

= `int (sin[(x + a) - 2a])/(sin(x + a))dx`

= `int (sin(x + a)cos2a - cos(x + a)sin2a)/(sin(x + a))dx`

= `int [(sin(x + a)cos2a)/(sin(x + a)) - (cos(x + a)sin2a)/(sin(x + a))]dx`

= `int [cos2a - sin 2a cot(x + a)]dx`

= `cos2a int 1dx - sin2a int cot(x + a)dx`

= x cos 2a – sin 2a log sin (x + a) + c

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