Notation for antiderivativeI came across the following: F ( x ) = ∫ x 3 cos ⁡ ( x ) d xwhere F is understood to be a primitive of x 3 cos ⁡ ( x ). I find this confusing, because of the "same" x appearing on both sides of the equality. To me, x is "integrated out" on the right side, and I prefer the notation: F ( x ) = ∫ 0 x u 3 cos ⁡ ( u ) d uor possibly: F = ∫ x 3 cos ⁡ ( x ) d xwithout mentioning the variable for F.

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Notation for antiderivativeI came across the following:  F  (  x  )  =  ∫      x    3    cos  ⁡  (  x  )  d  xwhere F is understood to be a primitive of       x    3    cos  ⁡  (  x  ). I find this confusing, because of the &quot;same&quot; x appearing on both sides of the equality. To me, x is &quot;integrated out&quot; on the right side, and I prefer the notation:  F  (  x  )  =      ∫          0              x            u    3    cos  ⁡  (  u  )  d  uor possibly:   F  =  ∫      x    3    cos  ⁡  (  x  )  d  xwithout mentioning the variable for F.

Kaydence Washington · Accepted Answer

Step 1The answer depends on what   ∫  f  (  x  )    d  x means, about which there is no universal agreement. One interpretation is: the set of all functions whose derivative is f. If this definition is accepted, then   = should really be read as   ∈. This is the same convenient abuse of notation as in             x      2        +    1    =  O  (  x  ). The other abuse is in writing F(x) when you mean F, and this is also convenient at times.So, this is how   F  ∈  ∫      x    3      d  x becomes   F  (  x  )  =  ∫      x    3      d  x.Notice that there is no integration involved in the above interpretation.Step 22nd interpretation: Someone may say that   ∫  f  (  x  )    d  x  is actually an integral, namely       ∫    a    x    f  (  t  )    d  t with unspecified a. If you subscribe to this point of view, then   sin  ⁡  x  =  ∫  cos  ⁡  x    d  x is a true statement while   sin  ⁡  x  +  5  =  ∫  cos  ⁡  x    d  x is false.

Arraryeldergox2 · Accepted Answer

Eplanation:It is indeed widely used, as is the uglier notation   F  (  x  )  =      ∫    x    f  (  w  )  d  w.   F  (  x  )  =  ∫  f  (  x  )  d  x means the family of primitives or antiderivatives: all F such that       F    ′    (  x  )  =  f  (  x  ).

Notation for antiderivative I came across the following: F ( x ) = ∫<!--

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