Evaluate the following integrals.
` int (sinx + cos x)/((sin x - cos x)^(3))dx `

Answer

To evaluate the integral

\[ \int \frac{\sin x + \cos x}{(\sin x - \cos x)^3} \, dx, \]

we will use substitution. Let's go through the steps:

### Step 1: Substitution
Let
\[ t = \sin x - \cos x. \]
Then, we differentiate \(t\) with respect to \(x\):
\[ \frac{dt}{dx} = \cos x + \sin x. \]
Thus,
\[ dt = (\cos x + \sin x) \, dx. \]

### Step 2: Rewrite the Integral
From our substitution, we can express \(\sin x + \cos x\) in terms of \(t\):
\[ \sin x + \cos x \, dx = dt. \]
Now, substituting \(t\) into the integral, we have:
\[ \int \frac{dt}{t^3}. \]

### Step 3: Integrate
The integral \(\int t^{-3} \, dt\) can be evaluated using the power rule for integration:
\[ \int t^{-3} \, dt = \frac{t^{-2}}{-2} + C = -\frac{1}{2t^2} + C. \]

### Step 4: Substitute Back
Now, we substitute back \(t = \sin x - \cos x\):
\[ -\frac{1}{2(\sin x - \cos x)^2} + C. \]

### Final Answer
Thus, the final result of the integral is:
\[ -\frac{1}{2(\sin x - \cos x)^2} + C. \]

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