Integrals Test ...

If $$2\displaystyle \int_{0}^{1}\tan^{-1}xdx=\displaystyle \int_{0}^{1}\cot^{-1}(1-x+x^{2})dx$$, then $$\displaystyle \int_{0}^{1}\tan^{-1}(1-x+x^{2})dx$$ is equal to: 

The value of $${\int }_{0}^{1}\dfrac{dx}{x+\sqrt{1-x^{2}}}$$ is

Let $$f: { (2,3)\rightarrow (0,1) }$$ be defined by $$f{ (x)=x-[x] }$$ then $$f^{ -1 }{ (x) }$$ equals

$$\displaystyle \int_{2}^{8}\dfrac {\sqrt {10-x}}{\sqrt {x}+\sqrt {10-x}}dx$$ is

$${\int}_{0}^{\pi}\dfrac{\sqrt{1-x}}{1+x}dx=$$

$$\int _ { 0 } ^ { \pi / 4 } \tan ^ { 2 } x d x$$ equals -

The value of 
$$\displaystyle\int _{ 5 }^{ 10 }{ \left( \sqrt { x+\sqrt { 20x-100 }  } +\sqrt { x-\sqrt { 20x-100 }  }  \right)  } dx$$ is 

Let $$f(x)= \left| \begin{matrix} \sec { x }  & \cos { x }  & \sec { ^{ 2 }x+\cot { x\csc { x }  }  }  \\ \cos { ^{ 2 }x }  & \cos { ^{ 2 }x }  & \csc { ^{ 2 }x }  \\ 1 & \cos { ^{ 2 }x }  & \cos { ^{ 2 }x }  \end{matrix} \right|$$. Then $${\int}_{0}^{\pi/2}f(x)dx$$ is equal to 

Evaluate
$$\int \dfrac{dx}{\sqrt{1-x}}$$

$$\displaystyle \overset{2\pi}{\underset{0}{\int}} x \,log \left(\dfrac{3 + \cos x}{3 - \cos x}\right)dx$$