Indefinite integral ∫ cos ⁡ 2 x sin 4 ⁡ x + cos 4 ⁡ x d x

Question

Indefinite integral   ∫            cos      ⁡      2      x                      sin        4            ⁡      x      +              cos        4            ⁡      x          d  x

Punktatsp · Accepted Answer

The reason why is that            (              sin        2            ⁡      (      x      )      +              cos        2            ⁡      (      x      )      )        2    =      sin    4    ⁡  (  x  )  +      cos    4    ⁡  (  x  )  +  2      sin    2    ⁡  (  x  )      cos    2    ⁡  (  x  )  =      1    2    =  1So,      sin    4    ⁡  (  x  )  +      cos    4    ⁡  (  x  )  =  1  −  2      sin    2    ⁡  (  x  )      cos    2    ⁡  (  x  )  =  1  −      1    2        sin    2    ⁡  (  2  x  )So  I  =  ∫            cos      ⁡      (      2      x      )                      sin        4            ⁡      (      x      )      +              cos        4            ⁡      (      x      )        d  x  =  ∫            cos      ⁡      (      2      x      )              1      −              1        2                    sin        2            ⁡      (      2      x      )        d  x

Boilanubjaini8f · Accepted Answer

HINT:      sin    4    ⁡  x  +      cos    4    ⁡  x  =  (      sin    2    ⁡  x  +      cos    2    ⁡  x      )    2    −  2      sin    2    ⁡  x      cos    2    ⁡  x  =  1  −                    (        sin        ⁡        2        x                  )          2                    2      Let   sin  ⁡  2  x  =  u

Indefinite integral ∫ cos ⁡ 2

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