Integral by use of substitution 3 cos ⁡ x d x 1 + 3 sin ⁡ x

Question

Integral by use of substitution             3      cos      ⁡      x      d      x              1      +      3      sin      ⁡      x

Eve Good · Accepted Answer

∫            3      cos      ⁡      (      x      )      d      x              1      +      3      sin      ⁡      (      x      )      Note that the derivative of   3  sin  ⁡  (  x  )  +  1 is present   [  3  cos  ⁡  (  x  )  ], so we can try to use u-substitution.  u  =  1  +  3  sin  ⁡  (  x  )  ← what&#039;s inside the     d  u  =  3  cos  ⁡  (  x  )  d  x  ← precisely our numeratorAfter substitution, we have  ∫            3      cos      ⁡      (      x      )      d      x              1      +      3      sin      ⁡      (      x      )        =  ∫      1          u        d  u  ∫            3      cos      ⁡      (      x      )      d      x              1      +      3      sin      ⁡      (      x      )        =  ∫      u          −              1        2              d  uIntegrating  ∫            3      cos      ⁡      (      x      )      d      x              1      +      3      sin      ⁡      (      x      )        =  2      u                  1        2              +  C  ∫            3      cos      ⁡      (      x      )      d      x              1      +      3      sin      ⁡      (      x      )        =  2      u    +  CWriting in terms of x  ∫            3      cos      ⁡      (      x      )      d      x              1      +      3      sin      ⁡      (      x      )        =  2      1    +    3    sin    ⁡    (    x    )    +  C

agantisbz · Accepted Answer

∫            3      cos      ⁡      x              1      +      3      sin      ⁡      x        d  x  =  ∫      1          u        d  utake   u  =  1  +  3  sin  ⁡  x  d  u  =  3  cos  ⁡  x  d  x  =  ∫      u          −      1              /            2        d  u  =      u          1              /            2        +  c  =      1    +    3    sin    ⁡    x    +  c

Integral by use of substitution (int 3cos x dx)/(sqrt(1+3sin x))

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