The equation of the tangent to the curve $y = \sqrt{9 - 3x^2}$ at the point where the ordinate and abscissa equal is...

If the line $y = 4x - 5$ is tangent to the curve $y^2 = ax^3 + b$ at the point $(2,3)$, then the value of $7a - 2b$ is...

If $\int\dfrac{\sin x}{\sin 4x}\, dx = \alpha\log\left|\dfrac{1 + \sin x}{1 - \sin x}\right| + \beta\log\left|\dfrac{1 + \sqrt{2}\sin x}{1 - \sqrt{2}\sin x}\right| + c$, then the value of $32(\alpha + \beta^2) =$

If $\int f(x)\, dx = g(x)$ then $\int x^3 f(x^2)\, dx$ is equal to