The value of $\int \frac{e^{x}(1+x) dx}{\cos^{2}(x e^{x})}$ is equal to

If $\int \cos ^{\frac{3}{5}} x \cdot \sin ^3 x \,d x = \frac{-1}{m} \cos ^{m} x + \frac{1}{n} \cos ^{n} x + c$,(where $c$ is the constant of integration),then $(m, n) = $

Let $f(x) = \frac{x}{(1 + x^7)^{1/7}}$ and $g(x) = (f \circ f \circ f \circ f \circ f \circ f \circ f)(x)$. Then $\int x^5 g(x) dx$ equals (where $C$ is the constant of integration):

If $\int \frac{\log \left(t+\sqrt{1+t^2}\right)}{\sqrt{1+t^2}} dt=\frac{1}{2}(g(t))^2+c$ where $c$ is a constant of integration,then $g(2)$ is equal to

$\int \frac{e^{\tan ^{-1} 2 x}}{1+4 x^2} dx =$

$\int \sin \sqrt{x} \, dx = $