Integrals Test ...

If $$I_n=\displaystyle\int^1_0\dfrac{dx}{(1+x^2)^n}; n\in N$$, then which of the following statements hold good?

  $$\int _ { 0 } ^ { 2 } \cfrac { \sqrt { x } } { \sqrt { x } + \sqrt { 2 - x } }$$ is equal to

$$\int _{ 0 }^{ \infty  }{ \frac { { x }^{ 2 }+1 }{ { x }^{ 4 }+{ 7x }^{ 2 }+1 }  } $$ dx=

The solution of $$x$$ of the equation $$\displaystyle \int_{\sqrt{2}}^{x}{\dfrac{dt}{t\sqrt{t^{2}-1}}}=\dfrac{\pi}{2}$$ is 

Find the value of the equation :  $$\int _ { \ln \lambda } ^ { \ln \left( \frac { 1 } { \lambda } \right) }  \dfrac { f \left( \dfrac { x ^ { 2 } } { 3 } \right) ( f ( x ) + f ( - x ) ) } { g \left( 3 x ^ { 2 } \right) ( g ( x ) - g ( - x ) ) } d x =?$$   where  $$\lambda > 1$$

Find the value of the equation  $$\int _ { 0 } ^ { \infty } \dfrac { d x } { \left( x + \sqrt { x ^ { 2 } + 1 } \right) ^ { 3 } } =?$$

The value of $$\displaystyle \int _{0}^{1}\dfrac{x^{4}(1-x)^{4}}{1+x^{2}}\ dx$$ is

If $$u _ { n } = \int _ { 0 } ^ { \pi / 2 } x ^ { n } \sin x \quad d x$$ then the value of $$u _ { 10 } + 90 \mathrm { u } _ { 8 }$$ is:

$$\int^{2/3}_0 \dfrac{dx}{4+9x^2}$$ equals 

$$\int _{ -1 }^{ 1/2 }{ \dfrac { { e }^{ x }\left( 2-{ x }^{ 2 } \right) dx }{ \left( 1-x \right) \sqrt { 1-{ x }^{ 2 } }  }  }$$ is equal to