The value of $$\int\limits_0^{\pi /2} {{{{{\sin }^3}x} \over {\sin x + \cos x}}dx} $$ is

Let $$f(x) = \int\limits_0^x {g(t)dt} $$ where g is a non-zero even function. If ƒ(x + 5) = g(x), then $$ \int\limits_0^x {f(t)dt} $$ equals-

If $$f(x) = {{2 - x\cos x} \over {2 + x\cos x}}$$ and g(x) = log_ex, (x > 0) then the value of integral

$$\int\limits_{ - {\pi \over 4}}^{{\pi \over 4}} {g\left( {f\left( x \right)} \right)dx{\rm{ }}} $$ is

$$\mathop {\lim }\limits_{x \to \infty } \left( {{n \over {{n^2} + {1^2}}} + {n \over {{n^2} + {2^2}}} + {n \over {{n^2} + {3^2}}} + ..... + {1 \over {5n}}} \right)$$ is equal to :