Definite integration of logarithmic function ∫ 0 π log ⁡ ( 5 − 4 sin ⁡ x ) d xI tried to proceed with this question using integration by parts but got stuck over ∫ 0 π ( − 4 x cos ⁡ x 5 − 4 sin ⁡ x ) d x

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Definite integration of logarithmic function      ∫    0          π        log  ⁡  (  5  −  4  sin  ⁡  x  )  d  xI tried to proceed with this question using integration by parts but got stuck over       ∫    0          π        (            −      4      x      cos      ⁡      x        5    −  4  sin  ⁡  x  )  d  x

Lohre1x · Accepted Answer

I think that you have some mistakes from the start. Using integartion by parts  ∫  ln  ⁡  (  5  −  4  sin  ⁡  (  x  )  )    d  x  =  x  log  ⁡  (  5  −  4  sin  ⁡  (  x  )  )  +  4  ∫            x      cos      ⁡      (      x      )              5      −      4      sin      ⁡      (      x      )          d  x      ∫    0    π    ln  ⁡  (  5  −  4  sin  ⁡  (  x  )  )    d  x  =  π  log  ⁡  (  5  )  +  4      ∫    0    π              x      cos      ⁡      (      x      )              5      −      4      sin      ⁡      (      x      )          d  xBy the way      ∫    0    π    ln  ⁡  (  5  −  4  sin  ⁡  (  x  )  )    d  x  =  2      ∫    0                  π        2              ln  ⁡  (  5  −  4  sin  ⁡  (  x  )  )    d  xwhich could approximate (make a plot of it) as the area of the two approximate triangles that is to say   ∼      π    2    ln  ⁡  (  5  )  =  2.53 while numerical integration would give   2.41

Definite integration of logarithmic function int^(pi)_0 log(5−4 sin x)dx

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