Question

\[ \int \frac{dx}{\sin x - \cos x + \sqrt{2}} = ? \]

• A. \( -\frac{1}{\sqrt{2}} \tan \left( \frac{x}{2} + \frac{\pi}{8} \right) + C \)
• B. \( \frac{1}{2} \tan \left( \frac{x}{2} + \frac{\pi}{8} \right) + C \)
• C. \( \frac{1}{\sqrt{2}} \cot \left( \frac{x}{2} + \frac{\pi}{8} \right) + C \)
• D. \( -\frac{1}{\sqrt{2}} \cot \left( \frac{x}{2} + \frac{\pi}{8} \right) + C \)

Hint:

To solve integrals of trigonometric expressions, use standard trigonometric identities and substitution techniques. Recognizing the structure of the integrand can greatly simplify the problem.

Answer 1

The Correct Option is C

We are asked to integrate the following expression: \[ I = \int \frac{dx}{\sin x - \cos x + \sqrt{2}}. \] Step 1: Simplifying the expression.
First, we simplify the expression by recognizing a standard trigonometric identity. Start by rewriting \( \sin x - \cos x \) in a form that makes it easier to integrate. We can express \( \sin x - \cos x \) as follows: \[ \sin x - \cos x = \sqrt{2} \left( \sin \left( x - \frac{\pi}{4} \right) \right). \] Thus, the original integral becomes: \[ I = \int \frac{dx}{\sqrt{2} \sin \left( x - \frac{\pi}{4} \right) + \sqrt{2}}. \]
Step 2: Substitution.
Now, factor out \( \sqrt{2} \) from the denominator: \[ I = \int \frac{dx}{\sqrt{2} \left( \sin \left( x - \frac{\pi}{4} \right) + 1 \right)}. \] Next, perform the substitution \( u = \frac{x}{2} + \frac{\pi}{8} \), which leads to \( du = \frac{dx}{2} \). This transforms the integral into: \[ I = \int \frac{2 du}{\sqrt{2} \cdot \sin \left( u \right)}. \]
Step 3: Applying standard integral.
The integral of the form \( \int \frac{du}{\sin u + 1} \) is a standard result: \[ I = \frac{1}{\sqrt{2}} \cot \left( u \right) + C. \] Substitute \( u = \frac{x}{2} + \frac{\pi}{8} \) back into the equation: \[ I = \frac{1}{\sqrt{2}} \cot \left( \frac{x}{2} + \frac{\pi}{8} \right) + C. \]
Step 4: Conclusion.
Thus, the integral evaluates to: \[ \boxed{\frac{1}{\sqrt{2}} \cot \left( \frac{x}{2} + \frac{\pi}{8} \right) + C}. \] Therefore, the correct answer is (c).