Integrals Test ...

The value of the definite integral $$\int _{ 2 }^{ 3 }{ \left[ \sqrt { 2x-\sqrt { 5\left( 4x-5 \right)  }  } +\sqrt { 2x+\sqrt { 5\left( 4x-5 \right)  }  }  \right]  } dx=$$ 

The integral $$\displaystyle \int_{\dfrac{\pi}{12}}^{\dfrac{\pi}{4}}{\dfrac{8\cos 2x}{(\tan x+\cot x)^{3}}dx}$$ equals :

Let $$f\left(x\right)+f\left(\dfrac{1}{x}\right)=F\left(x\right)$$ where $$f\left(x\right)=\displaystyle\int_{1}^{x}{\dfrac{\ln{t}}{1+t}dx}$$.Then $$F\left(e\right)=$$

The value of $$\displaystyle \int _{0}^{\infty}\dfrac{\sqrt{x^2+1}}{(x+\sqrt{x^{2}+1})^{n+1}} .dx  \ \forall \ n\ \in \ N- \{ \pm 1 \}$$ is

$$\displaystyle \int_{-3\pi}^{3\pi}{\sin^{2}\theta\sin^{2} 2\theta d \theta}$$ is equal to-

If $$P=\displaystyle \lim _{ n\rightarrow \infty  }{ \frac { { \left( \prod _{ r=1 }^{ n }{ \left( { n }^{ 3 }+{ r }^{ 3 } \right)  }  \right)  }^{ 1/n } }{ { n }^{ 3 } }  }$$  and $$\lambda =\displaystyle \int _{ 0 }^{ 1 }{ \frac { dx }{ 1+{ x }^{ 3 } }  } $$ then $$In P$$ is equal to

Let $$f:R\longrightarrow R,g : R\longrightarrow R$$ be continuous functions. then the value of integeral. 
$$\int _{ \ell n\lambda  }^{ \ell n/\lambda  }{ \frac { f\left( \dfrac { { x }^{ 2 } }{ 4 }  \right) \left[ f\left( x \right) -f\left( -x \right)  \right]  }{ g\left( \dfrac { { x }^{ 2 } }{ 4 }  \right) \left[ g\left( x \right) +g\left( -x \right)  \right]  }  } dx$$ is:

$$\displaystyle \int_{0}^{\pi/2}{(\sin x-\cos x).\log(\sin x+\cos x)dx}$$=

$$\int^2_0 (\sqrt{\dfrac{4-x}{x}} - \sqrt{\dfrac{x}{4-x}})dx$$ is equal to