Antiderivative of 1 4 sin ⁡ ( 2 x )I am trying to find the antiderivative of 1 4 sin ⁡ ( 2 x ) but i'mn ot sure how to find this. Can someone maybe give me what formula to use when it involves sin?

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Antiderivative of       1    4    sin  ⁡  (  2  x  )I am trying to find the antiderivative of       1    4    sin  ⁡  (  2  x  ) but i&#039;mn ot sure how to find this. Can someone maybe give me what formula to use when it involves sin?

Blaze Frank · Accepted Answer

Step 1The beauty of trig is that you can get wildly different looking results depending on how you start. For example take identify:   sin  ⁡  (  2  x  )  =  2  sin  ⁡  (  x  )  cos  ⁡  (  x  )Then you can write   ∫      1    4    sin  ⁡  (  2  x  )      d    x  =      1    2    ∫  sin  ⁡  (  x  )  cos  ⁡  (  x  )      d    x  .Set   u  =  cos  ⁡  (  x  ) then   d  u  =  −  sin  ⁡  (  x  )  d  x. So now we solve the indefinite integral:       1    2    ∫  −  u      d    u  =  −      1    4        u    2    +  CSubstituting back in for   u  =  cos  ⁡  (  x  ) we get  ∫      1    4    sin  ⁡  (  2  x  )      d    x  =  −      1    4        cos    2    ⁡  (  x  )  +  CThis looks very different but you can see that it&#039;s equivalent to the first answer given to your question by remembering the following:      cos    2    ⁡  (  x  )  =  1  −      sin    2    ⁡  (  x  )and       sin    2    ⁡  (  x  )  =            1      −      cos      ⁡      (      2      x      )        2   and finally keeping in mind that C of the two answers is not the same.

Arraryeldergox2 · Accepted Answer

Step 1  I  :=  ∫      1    4    sin  ⁡  (  2  x  )     d  x  I  =      1    4    ∫  sin  ⁡  (  2  x  )     d  xLet   u  =  2  x  ,  d  u  =  2     d  x    ⟹              d      u        2    =  d  x.  I  =      1    4    ⋅      1    2    ∫  sin  ⁡  (  u  )     d  u  I  =      1    8    ⋅  −  cos  ⁡  u  +  C  I  =  −      1    8    cos  ⁡  (  2  x  )  +  CWhy is this correct? We can take the derivative of our answer to see we get our original function back and that this and our primitive are equal up to a constant.      d          d      x            (    −          1      8        cos    ⁡    (    2    x    )    )    =  −      1    8    ⋅  −  sin  ⁡  (  2  x  )  ⋅  2  =      1    4    sin  ⁡  (  2  x  )It is helpful to note these few facts to answer your question about what to do when you have sin or cos in an integrand:  ∫  sin  ⁡  x     d  x  =  −  cos  ⁡  x  +  C  ∫  cos  ⁡  x     d  x  =  sin  ⁡  x  +  CNote that this just involves x, but you can still do your normal u-substitutions as necessary to get it in the form above, as I did.

Antiderivative of 1 4 </mfrac> sin ⁡ ( 2 x )

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