If $I_n = \int \tan^n x \ dx$,and $I_0 + I_1 + 2 I_2 + 2 I_3 + 2 I_4 + I_5 + I_6 = \sum_{K=1}^n \frac{\tan^K x}{K}$,then $n = $

Let $5 f(x)+4 f\left(\frac{1}{x}\right)=\frac{1}{x}+3$,where $x > 0$. Then $18 \int \limits_1^2 f(x) \, dx$ is equal to:

$\int (1+\tan^2 x)(1+2x \tan x) dx =$

$\int {\frac{{{{\sin }^{ - 1}}x - {{\cos }^{ - 1}}x}}{{{{\sin }^{ - 1}}x + {{\cos }^{ - 1}}x}}} dx = $

If $\int x^{49} \left[ \operatorname{Tan}^{-1} x^{50} + \frac{x^{50}}{1 + x^{100}} \right] dx = \frac{x^n}{k} f(x) + c$,then $f(x) - f\left(\sqrt[k]{x^n}\right) =$

The value of the integral $\int \frac{\sin \theta \cdot \sin 2 \theta \left(\sin ^{6} \theta+\sin ^{4} \theta+\sin ^{2} \theta\right) \sqrt{2 \sin ^{4} \theta+3 \sin ^{2} \theta+6}}{1-\cos 2 \theta} d \theta$ is (where $c$ is a constant of integration)