Let \( f(x) = \int x^{3}\sqrt{3 - x^{2}} \, dx \). If \( 5 f(\sqrt{2}) = -4 \), then \( f(1) \) is equal to:
\( -\frac{8\sqrt{2}}{5} \)
\( -\frac{4\sqrt{2}}{5} \)
\( -\frac{6\sqrt{2}}{5} \)
\( -\frac{2\sqrt{2}}{5} \)
If \( \displaystyle \int \left( \frac{1}{x} + \frac{1}{x^3} \right) \left( \sqrt[23]{3x^{-24} + x^{-26}} \right) dx = -\frac{\alpha}{3(\alpha + 1)} \left( 3x^\beta + x^\gamma \right)^{\frac{\alpha + 1}{\alpha}} + C, \quad x > 0, \, (\alpha, \beta, \gamma \in \mathbb{Z}) \), where \( C \) is the constant of integration, then \( \alpha+\beta+\gamma \) is equal to __________.
The integral \( \int_{0}^{\pi} \frac{8x \, dx}{4\cos^{2} x + \sin^{2} x} \) is equal to:
