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Question
Integrate the functions:
sin (ax + b) cos (ax + b)
Solution
Let `I = int sin (ax + b) cos (ax + b) dx`
Put sin (ax + b) = t
⇒ a cos (ax + b) dx = dt
∴ `I = 1/a int t dt = 1/a * t^2/2 + C`
`= 1/(2a) t^2 + C`
`= 1/ (2a) sin^2 (ax + b) + C`
Or, put cos (ax + b) = t
⇒ -a sin (ax + b) dx = dt
∴ `I = (-1)/a int dt = (-1)/a t^2/2 + C`
`= (-cos^2 (ax + b))/(2a) + C`
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