Question

\( \int \frac{dx}{\sin(x) + \cos(x)} = ? \)

Answer 1

To solve the integral \[ \int \frac{dx}{\sin(x) + \cos(x)} \] we can use the following steps:

1. Rewrite the denominator:
We can rewrite \( \sin(x) + \cos(x) \) as: \[ \sqrt{2} \left( \frac{1}{\sqrt{2}} \sin(x) + \frac{1}{\sqrt{2}} \cos(x) \right) \] Since \( \cos(\pi/4) = \frac{1}{\sqrt{2}} \) and \( \sin(\pi/4) = \frac{1}{\sqrt{2}} \), we can write: \[ \sin(x) + \cos(x) = \sqrt{2} \left( \cos(\pi/4) \sin(x) + \sin(\pi/4) \cos(x) \right) \] Using the identity \( \sin(a + b) = \sin(a) \cos(b) + \cos(a) \sin(b) \), we get: \[ \sin(x) + \cos(x) = \sqrt{2} \sin \left( x + \frac{\pi}{4} \right) \]

2. Substitute into the integral:
Now we substitute this expression into the original integral: \[ \int \frac{dx}{\sin(x) + \cos(x)} = \int \frac{dx}{\sqrt{2} \sin \left( x + \frac{\pi}{4} \right)} \]

3. Simplify and integrate:
Simplifying the integral: \[ \int \frac{dx}{\sqrt{2} \sin \left( x + \frac{\pi}{4} \right)} = \frac{1}{\sqrt{2}} \int \csc \left( x + \frac{\pi}{4} \right) dx \] We know that: \[ \int \csc(u) du = \ln | \csc(u) - \cot(u) | + C \] Therefore, the integral becomes: \[ \frac{1}{\sqrt{2}} \int \csc \left( x + \frac{\pi}{4} \right) dx = \frac{1}{\sqrt{2}} \ln \left| \csc \left( x + \frac{\pi}{4} \right) - \cot \left( x + \frac{\pi}{4} \right) \right| + C \]

Thus, the solution is:
\[ \int \frac{dx}{\sin(x) + \cos(x)} = \frac{1}{\sqrt{2}} \ln \left| \csc \left( x + \frac{\pi}{4} \right) - \cot \left( x + \frac{\pi}{4} \right) \right| + C \]