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

Consider the integral

$${I_m} = \int_0^\pi {{{\sin 2mx} \over {\sin x}}dx} $$, where m is a positive integer.

What is I_m equal to ?

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

Consider the integral

$${I_m} = \int_0^\pi {{{\sin 2mx} \over {\sin x}}dx} $$, where m is a positive integer.

Consider the following

I. I_m $$-$$ I_{m $$-$$ 1} is equal to 0

II. I_2m > I_m

Which of the above is/are correct?

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?