Let f(x) be a function such that $$f'\left( {{1 \over x}} \right) + {x^3}f'(x) = 0$$. What is $$\int_{ - 1}^1 {f(x)} \,dx$$ equal to ?

If $$\int_0^{\pi /2} {{{dx} \over {3\cos x + 5}} = k{{\cot }^{ - 1}}2} $$, then what is the value of k?

What is $$\int_1^3 {\left| {1 - {x^4}} \right|} \,dx$$ equal to ?

Read the following information carefully and answer the questions given below.

Consider the functions f(x) = xg(x) and g(x) = $$\left[ {{1 \over x}} \right]$$, where [ . ] is the greatest integer function.

What is $$\int_{1/3}^{1/2} {g(x)\,dx} $$ equal to ?