Competitive Thinking

Suppose that $F\left(x\right)$ is an anti-derivative of $f\left(x\right)=\frac{\sin x}{x}$, where $x>0$, then ${\int }_1^3\frac{\sin 2x}{x}dx$ can be expressed as

The value of integral ${\int }_{\frac{1}{\pi }}^{\frac{2}{\pi }}\frac{\sin \left(\frac{1}{x}\right)}{x^2}dx=$

The value of the integral ${\int }_0^{\frac{\pi }{4}}\frac{\sqrt{\tan x}}{\sin x\cos x}dx$ equals

If $I_n={\int }_0^{\frac{\pi }{4}}{\tan }^n\theta d\theta$, then $I_8+I_6$ equals

If $g\left(1\right)=g\left(2\right)$, then ${\int }_1^2{\left[fg\left(x\right)\right]}^{-1}{f}^{\text{'}}\left[g\left(x\right)\right]g^{\text{'}}\left(x\right)dx$ is equal to

If ${\int }_0^k\frac{dx}{2+18x^2}=\frac{\pi }{24}$, then the value of $k$ is

${\int }_{-1}^0\frac{dx}{x^2+2x+2}$ is equal to

The value of ${\int }_0^1\frac{x^2}{1+x^2}dx$ is