If $f(x) = \dfrac{\sin^{-1}x}{\sqrt{1 - x^2}}$ and $g(x) = e^{\sin^{-1}x}$, then the value of $\int f(x)g(x)\, dx = \ldots$

The function $f(x) = \int_0^x\dfrac{dt}{1 + \cos t}$ satisfies which of the following differential equations?

If $\int_a^b(x^2 - x)\, dx = 18, \int_a^b x^3\, dx = 0$, then $a + b$ is equal to

The value of the definite integral $\int_0^{\pi}\dfrac{1}{5 + 4\cos x}\, dx$ is equal to ...