Primitive of a function with sin ⁡ 1 x I have the next integral: ∫ ( sin ⁡ 1 x x 2 ( 4 + 3 sin ⁡ 2 x ) ) d x , x ∈ ( 0 , ∞ ) I used the substitution u = 1 x and I got − ∫ ( sin ⁡ u ( 4 + 3 sin ⁡ 2 u ) ) d uCan somebody give me some tips about what should I do next, please?

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Primitive of a function with   sin  ⁡      1    x  I have the next integral:  ∫            (                          sin        ⁡                  1          x                                      x          2                          (          4          +          3          sin          ⁡                      2            x                    )                                )          d  x  ,    x  ∈            (        0  ,  ∞            )      I used the substitution   u  =      1    x   and I got  −  ∫            (                          sin        ⁡        u                    (        4        +        3        sin        ⁡        2        u        )                        )          d  uCan somebody give me some tips about what should I do next, please?

enfujahl · Accepted Answer

The substitution u=1/x yields   d  x  =  −      1          u      2          d  u, so the integral becomes  ∫            −      sin      ⁡      u              4      +      3      sin      ⁡      2      u          d  u  =  ∫            −      sin      ⁡      u              4      +      3      sin      ⁡      2      u          d  u This can be improved by setting   u  =  π      /    4  −  v, so we get      1          2        ∫            cos      ⁡      v      −      sin      ⁡      v              4      +      3      cos      ⁡      2      v          d  v  =      1          2                  (        ∫            cos      ⁡      v              7      −      6              sin        2            ⁡      v          d  v  −  ∫            sin      ⁡      v              6              cos        2            ⁡      v      +      1          d  v            )      that you should be able to manage.

Primitive of a function with sin ⁡ 1 x </mfrac> I have the next

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