The line $x + y = 0$ touches the curve $y^2 = ax^3 + b$ at $(1, -1)$ then values of $a$ and $b$ respectively are ...........

The function $f(x) = x(x + 3)e^{-\left(\frac{1}{2}\right)x}$ satisfies all the conditions of Rolle's theorem in $[-3, 0]$, then $c =$

If $\int f(x)dx = g(x) + c$ then $\int f^{-1}(x)dx =$

$\int\dfrac{\log x}{(1 + \log x)^2}dx =$