Let's do some calculus! (1)

Calculus Level 5

csc2xcot2x1dx2sinx12sin2xdx\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 1010.
  • 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 5,17,235,17,23, then the answer would be log10(5×17×23)\log_{10}{\left(5\times17\times23\right)} or log105+log1017+log1023\log_{10}{5}+\log_{10}{17}+\log_{10}{23} (whichever way necessary) which would be approximately equal to 3.2913.291.
  • Give your answer correct to 3 decimal places.

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