Let's do some calculus! (1)

Calculus Level 5

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

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

$$\begin{array} {} & \displaystyle \int{\sqrt{\csc^{2}x-2}}\,dx & \implies (2) \\ & \displaystyle \int{\dfrac{\sqrt{\cos 2x}}{2\sin x}}\,dx & \implies (3) \\ & \displaystyle \int{\dfrac{\sqrt{\cos 2x}}{\sin x}}\,dx & \implies (5) \\ & \displaystyle \int{\sqrt{\csc^{2}x-1}}\,dx & \implies (7) \end{array}$$

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 $3.291$.
• Give your answer correct to 3 decimal places.

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