Question

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

Hint:

For definite integrals involving trigonometric terms, simplify the denominator and use substitution for ease.

Answer 1

1. Simplify the denominator: Use the identity: \[ \sin x + \cos x = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right). \] Thus: \[ \frac{1}{\sin x + \cos x} = \frac{1}{\sqrt{2} \sin\left(x + \frac{\pi}{4}\right)}. \] 2. Rewrite the integral: \[ \int_{0}^{\pi/4} \frac{1}{\sin x + \cos x} \, dx = \frac{1}{\sqrt{2}} \int_{0}^{\pi/4} \csc\left(x + \frac{\pi}{4}\right) \, dx. \] 3. Substitute: Let \( u = x + \frac{\pi}{4} \), so \( du = dx \). Change the limits: \[ u = \frac{\pi}{4} \to \frac{\pi}{2}, \quad u = 0 \to \frac{\pi}{4}. \] The integral becomes: \[ \frac{1}{\sqrt{2}} \int_{\pi/4}^{\pi/2} \csc u \, du. \] 4. Integrate \( \csc u \): \[ \int \csc u \, du = \ln|\csc u - \cot u|. \] 5. Evaluate at the limits: \[ \frac{1}{\sqrt{2}} \left[\ln|\csc(\pi/2) - \cot(\pi/2)| - \ln|\csc(\pi/4) - \cot(\pi/4)|\right]. \] Substitute values: \[ \csc(\pi/2) = 1, \, \cot(\pi/2) = 0, \, \csc(\pi/4) = \sqrt{2}, \, \cot(\pi/4) = 1. \] Simplify: \[ \ln|\csc(\pi/2) - \cot(\pi/2)| = \ln(1), \quad \ln|\csc(\pi/4) - \cot(\pi/4)| = \ln(\sqrt{2} - 1). \] Final result: \[ \frac{1}{\sqrt{2}} \left[0 - \ln(\sqrt{2} - 1)\right] = -\frac{\ln(\sqrt{2} - 1)}{\sqrt{2}}. \] Final Answer: \[ \boxed{-\frac{\ln(\sqrt{2} - 1)}{\sqrt{2}}.} \]