Let's do some calculus! (1)

Calculus Level 4

$\displaystyle \int{\dfrac{\csc^{2}x}{\sqrt{\cot^{2}x-1}}}\,dx - 2\int{\dfrac{\sin x}{\sqrt{1-2\sin^{2}x}}}\,dx$

Which of the given integrating functions is equivalent to the integration above:

$$A) \displaystyle \int{\sqrt{\csc^{2}x-2}}\,dx ~~~\cdots (2) \\ B) \displaystyle \int{\dfrac{\sqrt{\cos 2x}}{2\sin x}}\,dx ~~~\cdots (3) \\ C) \displaystyle \int{\dfrac{\sqrt{\cos 2x}}{\sin x}}\,dx ~~~\cdots (5) \\ D) \displaystyle \int{\sqrt{\csc^{2}x-1}}\,dx ~~~\cdots (7)$$

Details:

• Multiple options may be correct.
• Find out the answer as the product of the numbers corresponding to the correct integrating function(s), then type in your answer as the logarithm of the product at base $$10$$.
• The corresponding numbers are the prime numbers in parentheses at the end of each option.
• Example: If there were three correct answers with corresponding numbers $$5,17,23$$, then the answer would be $$\log_{10}{\left(5\times17\times23\right)}$$ or $$\log_{10}{5}+\log_{10}{17}+\log_{10}{23}$$ (whichever way necessary) which would be approximately equal to $$3.291$$.