Find the integral: ∫ cos 2 ⁡ ( 2 x ) d x

Question

Find the integral:  ∫      cos          2        ⁡  (  2  x  )  d  x

fobiosofia3ql · Accepted Answer

Use Integration by Substitution.Let   u  =  2  x  ,  d  u  =  2  d  x,then   d  x  =      1    2    d  uUsing   u and   du above, rewrite   ∫      cos          2        ⁡  (  2  x  )  d  x  ∫                    cos                  2                    ⁡      u        2    d  uUse Constant Factor Rule:   ∫  c  f  (  x  )  d  x  =  c  ∫  f  (  x  )  d  x  ..      1    2    ∫      cos          2        ⁡  u  d  uUse Pythagorean Identities:       cos          2        ⁡  x  =      1    2    +            cos      ⁡      2      x        2        1    2    ∫      1    2    +            cos      ⁡      2      u        2    d  uUse Sum Rule:  f  (  x  )  +  g  (  x  )  d  x  =  ∫  f  (  x  )  d  x  +  ∫  g  (  x  )  d  x  .      1    2    (  ∫      1    2    d  u  +  ∫            cos      ⁡      2      u        2    d  u  )Use this rule:   ∫  a  d  x  =  a  x  +  C      1    2    (      u    2    +  ∫            cos      ⁡      2      u        2    d  u  )Use Constant Factor Rule:   ∫  c  f  (  x  )  d  x  =  c  ∫  f  (  x  )  d  x      1    2    (      u    2    +      1    2    ∫  cos  ⁡  2  u  d  uUse Integration by Substitution on   ∫  c  o  s  2  u  d  u.Let   w  =  2  u  ,  d  w  =  2  d  u,  then   d  u  =      1    2    d  wUsing   w and   dw above, rewrite   ∫  cos  ⁡  2  u  d  u  ∫            cos      ⁡      w        2    d  wUse Constant Factor Rule:  ∫  c  f  (  x  )  d  x  =  c  ∫  f  (  x  )  d  x  ..      1    2    ∫  cos  ⁡  w  d  wUse Trigonometric Integration: the integral of   cos  ⁡  w is   sin  ⁡  w            sin      ⁡      w        2  Substitute   w  =  2  u back into the original integral.            sin      ⁡      2      u        2  Rewrite the integral with the completed substitution.      u    4    +            sin      ⁡      2      u        8  Substitute   u  =  2  x back into the original integral.            2      x        4    +            sin      ⁡      (      2      ×      2      x      )        8  Add constant.      x    2    +            sin      ⁡      4      x        8    +  C

Find the integral: int cos^(2) (2x)dx

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