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We'll determine the indefinite integral by changing the variable.
We'll note the denominator sin x - cos x = t(x)
We'll differentiate the denominator:
(sin x - cos x)' = [cos x - (-sin x)]dx
(cos x + sin x)dx = dt
We'll notice that the numerator of the function is the result of differentiating the function.
We'll calculate the integral:
Int f(x) = Int dt/t
Int dt/t = ln |t| + C
But t = sin x - cos x
Int f(x) = ln|sin x - cos x| + C
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We have to find the indefinite integral of f(x) = (sin x + cos x)/(sin x - cos x).
First let us express sin x - cos x as t.
So t = sin x - cos x
f(x) = (sin x + cos x) / t
Now dt / dx = cos x + sin x
Or dt = (cos x + sin x) dx
Now f(x) dx = dt / t
Int [ f(x) dx ] = Int [ dt / t ] = ln | t | + C
=> ln | sin x - cos x |
So the required integral is ln|sin x - cos x| + C
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