How to evaluate the integral ∫ 1 + sin ⁡ ( x ) d x

Question

How to evaluate the integral   ∫      1    +    sin    ⁡    (    x    )    d  x

aniizl · Accepted Answer

(  1  +  sin  ⁡  x  )  =            (      cos      ⁡                        x          2                    +      sin      ⁡                        x          2                    )        2      ⟹        1    +    sin    ⁡    x    =      |    cos    ⁡                  x        2              +    sin    ⁡                  x        2              |

Nasir Kim · Accepted Answer

In the first quadrant,      1    +    sin    ⁡    x    =                    1        −                  sin          2                ⁡        x                    1        −        sin        ⁡        x              =  ±                    sin        ′            ⁡      x              1      −      sin      ⁡      x      which you can integrate mentally, giving   ∓  2      1    −    sin    ⁡    x

How to evaluate the integral ∫

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