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The value of √2 ∫ [sin x dx] / [sin(x - π/4)] is

1. x + log|cos(x - π/4)| + c
2. x - log|sin(x - π/4)| + c
3. x + log|sin(x - π/4)| + c
4. x - log|cos(x - π/4)| + c

Correct answer: x + log|sin(x - π/4)| + c

Solution

The integrand becomes 1+(sinx+cosx)/(sinx-cosx). Integrating with u=sinx-cosx gives x+log|sinx-cosx|+c = x+log|sin(x-pi/4)|+c (constant absorbed).

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