If $f(x) = \dfrac{1}{\log x}$ and $g(x) = \dfrac{1}{(\log x)^2}$, then the value of $\displaystyle\int [f(x) - g(x)]\,dx$ is...

If $\displaystyle\int \dfrac{dx}{x^{7/2}(x^4 + 1)^{3/8}} = m\left(\dfrac{x^4 + 1}{x^4}\right)^n + c$, where $c$ is a constant of integration, then the value of $\dfrac{n}{m}$ is...

If $\int f'(x) \cdot e^{x^2}\,dx = (x - 1) \cdot e^{x^2} + k$, where $k$ is constant of integration, then $f(x) = \ldots$

If $\int e^{x + \tan^{-1}x}\left(\dfrac{x^2 + 2}{\sec^2(\tan^{-1}x)}\right)dx = e^{f(x)} + c$, then $\ldots$