Antiderivative of piecewise functionI want to calculate the anti-derivative of: f ( x ) = { − sin ⁡ ( x ) if x ≥ 0 1 − x 2 if x &lt; 0 such as F ( π / 2 ) = 0I calculated both antiderivatives, thus I did get cos(x) for the first one and x − x 3 / 3. Question is to calculate F ( π ) + F ( − 1 ). But If I calculate that with null constant, I get − 5 / 3 which is wrong.Then could you help me on which value should I give to the constants C 1 + C 2 ?

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Antiderivative of piecewise functionI want to calculate the anti-derivative of:  f  (  x  )  =      {                            −          sin          ⁡          (          x          )                                      if                     x          ≥          0                                      1          −                      x            2                                                if                     x          &amp;lt;          0                        such as   F  (  π      /    2  )  =  0I calculated both antiderivatives, thus I did get cos(x) for the first one and   x  −      x    3        /    3. Question is to calculate   F  (  π  ) +   F  (  −  1  ). But If I calculate that with null constant, I get   −  5      /    3 which is wrong.Then could you help me on which value should I give to the constants       C    1    +      C    2  ?

Superina0xb4i · Accepted Answer

Step 1The question, as you stated it, is ill-posed. The anti-derivative of a piecewise continuous function like f is defined up to a constant on each &quot;piece&quot;. Thus, the anti-derivatives of f are  F  (  x  )  =      {                            cos          ⁡          (          x          )          +                      C            1                                                if                     x          ≥          0                                      x          −                                    x              3                        3                    +                      C            2                                                if                     x          &amp;lt;          0                        Step 2With the condition that   F  (  π      /    2  )  =  0, we can solve for       C    1  , but we do not have enough information left to solve for       C    2  . Presumably, the question is implicitly assuming something; if I had to guess, it&#039;s assuming that F(x) is continuous. If so, then you can use the equation       lim          x      →              0        +              F  (  x  )  =      lim          x      →              0        −              F  (  x  ) to solve for       C    2  .

anniferathonniz8km · Accepted Answer

Explanation:As you calculated that antiderivative for   x  ⩾  0 is   cos  ⁡  x  +      c    1   and for   x  &amp;lt;  0 it is   x  −            x      3        3    +      c    2  . As antiderivative should be continuous we have       c    1    +  1  =      c    2  , so we obtain   F  (  x  )  =      {                            cos          ⁡          (          x          )          +          c                                      if                     x          ≥          0                                      x          −                                    x              3                        3                    +          c          +          1                                      if                     x          &amp;lt;          0                        now you can get c from condition   F  (  π      /    2  )  =  0

Antiderivative of piecewise function I want to calculate the anti-derivative of: f ( x

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