Find the integral: ∫ csc 3 ⁡ x d x

Question

Find the integral:  ∫      csc          3        ⁡  x  d  x

flasheadita237m · Accepted Answer

Use Integration by Parts on   ∫      csc          3        ⁡  x  d  xLet   u  =  csc  ⁡  x  ,  d  v  =      csc          2        ⁡  x  ,  d  u  =  −  csc  ⁡  x  cot  ⁡  x  d  x  ,  v  =  −  cot  ⁡  xSubstitute the above into   u  v  −  ∫  v  d  u  .  −  csc  ⁡  x  cot  ⁡  x  −  ∫      cot          2        ⁡  csc  ⁡  x  d  xUse Pythagorean Identities:       cot          2        ⁡  x  =      csc          2        ⁡  x  −  1  −  csc  ⁡  x  cot  ⁡  x  −  ∫  (      csc          2        ⁡  x  −  1  )  csc  ⁡  x  d  xExpand.  −  csc  ⁡  x  cot  ⁡  x  −  ∫      csc          3        ⁡  x  −  csc  ⁡  x  d  xUse Sum Rule:   ∫  f  (  x  )  +  g  (  x  )  d  x  =  ∫  f  (  x  )  d  x  +  ∫  g  (  x  )  d  x  −  csc  ⁡  x  cot  ⁡  x  −  ∫      csc          3        ⁡  x  d  x  +  ∫  csc  ⁡  x  d  xSet it as equal to the original integral   ∫      csc          3        ⁡  x  d  x  ∫      csc          3        ⁡  x  d  x  =  −  csc  ⁡  x  cot  ⁡  x  −  ∫      csc          3        ⁡  x  d  x  +  ∫  csc  ⁡  x  d  xAdd   ∫      csc          3        ⁡  x  d  x to both sides.  ∫      csc          3        ⁡  x  d  x  +  ∫      csc          3        ⁡  x  d  x  =  −  csc  ⁡  x  cot  ⁡  x  +  ∫  csc  ⁡  x  d  xSimplify   ∫      csc          3        ⁡  x  d  x  +  ∫      csc          3        ⁡  x  d  x to   2  ∫      csc          3        ⁡  x  d  x  2  ∫      csc          3        ⁡  x  d  x  =  −  csc  ⁡  x  cot  ⁡  x  +  ∫  csc  ⁡  x  d  xDivide both sides by   2  ∫      csc          3        ⁡  x  d  x  =            −      csc      ⁡      x      cot      ⁡      x      +      ∫      csc      ⁡      x      d      x        2  Original integral solved.            −      csc      ⁡      x      cot      ⁡      x      +      ∫      csc      ⁡      x      d      x        2  Use Trigonometric Integration: the integral of   csc  ⁡  x is   ln  ⁡  (  csc  ⁡  x  −  cot  ⁡  x  )            −      csc      ⁡      x      cot      ⁡      x      +      ln      ⁡      (      csc      ⁡      x      −      cot      ⁡      x      )        2  Add constant.            −      csc      ⁡      x      cot      ⁡      x      +      ln      ⁡      (      csc      ⁡      x      −      cot      ⁡      x      )        2    +  C

Find the integral: int csc^3 xdx

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