Definite Integration

Let $$f:[0,\infty ) \to [0,\infty )$$ be defined as $$f(x) = \int_0^x {[y]dy}$$ where [x] is the greatest integer less than or equal to x. Which of the following is true?

If f(x) = \left\{ {\matrix{ {\int\limits_0^x {\left( {5 + \left| {1 - t} \right|} \right)dt,} } & {x > 2} \cr {5x + 1,} & {x \le 2} \cr } } \right., then

The value of the integral $$\int\limits_{ - 1}^1 {\log \left( {x + \sqrt {{x^2} + 1} } \right)dx}$$ is :

The value of the definite integral $$\int\limits_{ - {\pi \over 4}}^{{\pi \over 4}} {{{dx} \over {(1 + {e^{x\cos x}})({{\sin }^4}x + {{\cos }^4}x)}}}$$ is equal to :

The value of $$\int\limits_{{{ - 1} \over {\sqrt 2 }}}^{{1 \over {\sqrt 2 }}} {{{\left( {{{\left( {{{x + 1} \over {x - 1}}} \right)}^2} + {{\left( {{{x - 1} \over {x + 1}}} \right)}^2} - 2} \right)}^{{1 \over 2}}}dx}$$ is :

If the value of the integral $$\int\limits_0^5 {{{x + [x]} \over {{e^{x - [x]}}}}dx = \alpha {e^{ - 1}} + \beta }$$, where $$\alpha$$, $$\beta$$ $$\in$$ R, 5$$\alpha$$ + 6$$\beta$$ = 0, and [x] denotes the greatest integer less than or equal to x; then the value of ($$\alpha$$ + $$\beta$$)² is equal to :

The value of $$\int\limits_{ - {\pi \over 2}}^{{\pi \over 2}} {\left( {{{1 + {{\sin }^2}x} \over {1 + {\pi ^{\sin x}}}}} \right)} \,dx$$ is

$$\int\limits_6^{16} {{{{{\log }_e}{x^2}} \over {{{\log }_e}{x^2} + {{\log }_e}({x^2} - 44x + 484)}}dx}$$ is equal to :

The value of the integral $$\int\limits_0^1 {{{\sqrt x dx} \over {(1 + x)(1 + 3x)(3 + x)}}}$$ is :

If [x] is the greatest integer $$\le$$ x, then $${\pi ^2}\int\limits_0^2 {\left( {\sin {{\pi x} \over 2}} \right)(x - [x]} {)^{[x]}}dx$$ is equal to :