The slope of the tangent to the curve $y = \int_0^x \frac{1}{1+t^3} dt$ at the point where $x = 1$ is

If the normal to the curve $x^{2/3} + y^{2/3} = a^{2/3}$ makes an angle $\phi$ with the $X$-axis,then the equation of that normal is

The sum of the intercepts on the coordinate axes made by the tangent to the curve $\sqrt{x} + \sqrt{y} = \sqrt{a}$ is

The tangent to the curve $y = \frac{1}{x^2 + 2x + 5}$ which is parallel to the $X$-axis is:

Find the equation of the normal lines to the curve $3x^{2}-y^{2}=8$ which are parallel to the line $x+3y=4$.

The perpendicular distance from the origin to the normal drawn at any point on the curve $x=a(\cos \theta+\theta \sin \theta), y=a(\sin \theta-\theta \cos \theta)$ is