What will be the value of this integral ∫ 0 π 2 [ e ln ⁡ ( cos ⁡ x ⋅ d ( cos ⁡ x ) d x ) ] d x We know that a l o g a ( c ) = c .But in this question, the expression in logarithm function evaluates to be negative, for which log is not defined. ∫ 0 π 2 [ e ln ⁡ ( − s i n ( 2 x ) 2 ) ] d x So this means we cannot apply the log property? my working: = ∫ 0 π 2 [ cos ⁡ x ⋅ ( − sin ⁡ x ) ] d x = [ cos 2 ⁡ x 2 ] 0 π 2 = − 1 2 Is this integral, with undefined log, correct?

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What will be the value of this integral                    ∫                  0                                                                    π                2                                                                [                  e                      ln            ⁡                          (              cos              ⁡              x              ⋅                                                d                  (                  cos                  ⁡                  x                  )                                                  d                  x                                            )                                      ]            d      x      We know that                     a                  l          o                      g            a                    (          c          )                    =      c      .But in this question, the expression in logarithm function evaluates to be negative, for which log is not defined.                    ∫                  0                                                                    π                2                                                                [                  e                      ln            ⁡                          (              −                                                s                  i                  n                  (                  2                  x                  )                                2                            )                                      ]            d      x      So this means we cannot apply the log property? my working:            =              ∫                  0                                                                    π                2                                                                [        cos        ⁡        x        ⋅        (        −        sin        ⁡        x        )        ]              d    x    =            [                                                  cos              2                        ⁡            x                    2                    ]              0                      π        2              =                              −                                    1              2                                          Is this integral, with undefined log, correct?

postotnojeyf · Accepted Answer

The question is: Which logarithm function are you using?If you are using the real-valued   ln  :  (  0  ,  +  ∞  )  →      R  , your integrand is simply not defined on the interval             [        0  ,      π    2              ]      , and hence the integral neither.I put it into Alpha and it gave me -1/2 as answer.Unless you tell it not to, Wolfram Alpha happily uses a branch of the complex logarithm, and you only have definition problems for the integrand at   0 and       π    2  , where   cos  ⁡  x  =  0 or       d          d      x        cos  ⁡  x  =  0. At all other points, the integrand is well-defined - the ambiguity of   ln is cancelled by the periodicity of   exp, so we don&#039;t even need to choose a consistent branch of the logarithm, we might choose a branch for any point independently of the other choices. Then you have, by definition of the logarithm  exp  ⁡            (        ln  ⁡            (        cos  ⁡  x            d      (      cos      ⁡      x      )              d      x                  )                  )        =  cos  ⁡  x            d      (      cos      ⁡      x      )              d      x        =      1    2    ⋅            d      (              cos        2            ⁡      x      )              d      x        ,and the integral evaluates to   −      1    2   regardless of how we define the integrand at the endpoints of the interval.

Ayanna Goodman · Accepted Answer

Even integrals can be undefined. In your case there will be no curve drawn for  f  (  x  )  =      e          −              1        2            s      i      n      2      x      so the area is also undefined.

What will be the value of this integral int_(0)^(((pi)/(2)))[ e^(ln(cos x * (d( cos x))/(dx))) ]dx We know that a^(log_a(c)) = c

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