Antiderivative for a function for integrationI have: f ( x ) = cos ⁡ ( x ) × e sin ⁡ ( x ) and the fitting Antiderivative: F ( x ) = e sin ⁡ ( x ) Can someone please explain to me how I get from f(x) to F(x)? In small steps?

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Antiderivative for a function for integrationI have:   f  (  x  )  =  cos  ⁡  (  x  )  ×      e          sin      ⁡      (      x      )       and the fitting Antiderivative:  F  (  x  )  =      e          sin      ⁡      (      x      )      Can someone please explain to me how I get from f(x) to F(x)? In small steps?

ladaroh · Accepted Answer

Explanation:Put   y  =  s  i  n  (  x  ), then   d  y  =  c  o  s  (  x  )  d  x,   ∫      e    y    d  y  =      e    y    =      e          (      s      i      n      (      x      )      )

spockmonkey40 · Accepted Answer

Step 1Just, substitute   u  =  sin  ⁡  (  x  ) and   d  u  =  cos  ⁡  (  x  )     d  x and use that the integral of       e    x   equals       e    x   plus a integration constant.Step 2  f  (  x  )  =  cos  ⁡  (  x  )      e          sin      ⁡      (      x      )        →  F  (  x  )  =  ∫  cos  ⁡  (  x  )      e          sin      ⁡      (      x      )           d  x  =  ∫      e          u           d  u  =      e    u    +  C  =      e          sin      ⁡      (      x      )        +  C

Antiderivative for a function for integration I have: f ( x ) = cos ⁡<

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