Step 1. Evaluate the integral:
\(\int_{\frac{\pi}{8}}^{\frac{\pi}{3}} \sqrt{1 - \sin 2x} \, dx\)
Step 2. Rewrite \( \sqrt{1 - \sin 2x} \) using trigonometric identities:
\(\int_{\frac{\pi}{8}}^{\frac{\pi}{3}} |\sin x - \cos x| \, dx\)
Step 3. Split the integral based on the intervals where \( \sin x - \cos x \) changes sign:
\(= \int_{\frac{\pi}{8}}^{\frac{\pi}{4}} (\cos x - \sin x) \, dx + \int_{\frac{\pi}{4}}^{\frac{\pi}{3}} (\sin x - \cos x) \, dx\)
Step 4. Solve each integral:
\(= -1 + 2\sqrt{2} - \sqrt{3}\)
Thus, we have:
\(\alpha + \beta \sqrt{2} + \gamma \sqrt{3} = -1 + 2\sqrt{2} - \sqrt{3}\)
where \( \alpha = -1 \), \( \beta = 2 \), and \( \gamma = -1 \).
Step 5. Calculate \( 3\alpha + 4\beta - \gamma \):
\(3\alpha + 4\beta - \gamma = 3(-1) + 4(2) - (-1) = -3 + 8 + 1 = 6\)
The Correct Answer is: \( 3\alpha + 4\beta - \gamma = 6 \).
The value of \[ \lim_{x \to \infty} \left( x - \sqrt{x^2 + x} \right) \] is equal to:
Integration of \(\ln(x)\) with \(x\), i.e. \(\int \ln(x)dx =\) __________.
If \[ \frac{dy}{dx} + 2y \sec^2 x = 2 \sec^2 x + 3 \tan x \cdot \sec^2 x \] and
and \( f(0) = \frac{5}{4} \), then the value of \[ 12 \left( y \left( \frac{\pi}{4} \right) - \frac{1}{e^2} \right) \] equals to: