Question

Evaluate: \[ \int_{0}^{\pi/4} \frac{1}{\sin x + \cos x} \, dx. \]

Hint:

To evaluate integrals involving \( \sin x + \cos x \), simplify using trigonometric identities, such as expressing the sum as a single sine or cosine term.

Answer 1

Step 1: Simplify the integrand
The given integral is: \[ I = \int_{0}^{\pi/4} \frac{1}{\sin x + \cos x} \, dx. \] Use the identity \( \sin x + \cos x = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) \). Substituting: \[ I = \int_{0}^{\pi/4} \frac{1}{\sqrt{2} \sin\left(x + \frac{\pi}{4}\right)} \, dx = \frac{1}{\sqrt{2}} \int_{0}^{\pi/4} \csc\left(x + \frac{\pi}{4}\right) \, dx. \] Step 2: Integrate \( \csc(x + \pi/4) \)
The integral of \( \csc(x) \) is: \[ \int \csc x \, dx = \log|\csc x - \cot x| + C. \] Substituting this, we have: \[ I = \frac{1}{\sqrt{2}} \left[\log\left|\csc\left(x + \frac{\pi}{4}\right) - \cot\left(x + \frac{\pi}{4}\right)\right|\right]_{0}^{\pi/4}. \] Step 3: Evaluate the limits
At \( x = \pi/4 \): \[ \csc\left(\frac{\pi}{4} + \frac{\pi}{4}\right) = \csc\left(\frac{\pi}{2}\right) = 1, \quad \cot\left(\frac{\pi}{4} + \frac{\pi}{4}\right) = \cot\left(\frac{\pi}{2}\right) = 0. \] At \( x = 0 \): \[ \csc\left(0 + \frac{\pi}{4}\right) = \csc\left(\frac{\pi}{4}\right) = \sqrt{2}, \quad \cot\left(0 + \frac{\pi}{4}\right) = \cot\left(\frac{\pi}{4}\right) = 1. \] Substitute these values: \[ I = \frac{1}{\sqrt{2}} \left[\log\left(1 - 0\right) - \log\left(\sqrt{2} - 1\right)\right] = \frac{1}{\sqrt{2}} \left[\log(1) - \log(\sqrt{2} - 1)\right]. \] Step 4: Simplify the result
\[ I = \frac{1}{\sqrt{2}} \log(\sqrt{2} + 1) - \frac{1}{\sqrt{2}} \log(\sqrt{2} - 1). \]