$\int_1^e \frac{1}{x} \, dx$ is equal to

The value of $\int_{1/e}^{\tan x} \frac{t \, dt}{1 + t^2} + \int_{1/e}^{\cot x} \frac{dt}{t(1 + t^2)} = $

If $\int_{1}^{2} \frac{dx}{(x^2 - 2x + 4)^{3/2}} = \frac{k}{k+5}$,then $k$ is equal to

$\int_0^1 \log (x+1) \, dx =$

The integral $80 \int_0^{\frac{\pi}{4}} \left( \frac{\sin \theta + \cos \theta}{9 + 16 \sin 2 \theta} \right) d \theta$ is equal to :

For a real number $x$,let $[x]$ denote the greatest integer less than or equal to $x$. The smallest positive integer $n$ for which the integral $\int_{1}^{n} [x][\sqrt{x}] \, dx$ exceeds $60$ is