Find the derivative of an integral: d d x ∫ π 2 x 3 cos ⁡ ( t ) d t

Question

Find the derivative of an integral:      d          d      x            ∫                  π        2                            x        3              cos  ⁡  (  t  )  d  t

naudiliwnw · Accepted Answer

Substitute   u for       x    3   :      d          d      x            ∫                  π        2                    u        cos  ⁡  (  t  )  d  tWe’ll use the chain rule to find the derivative, because we want to transform the integral into a form that works with the second fundamental theorem of calculus:      d          d      u            (          ∫                        π          2                            u              cos    ⁡          (      t      )        d    t    )    ×            d      u              d      x      Now we can find the derivative by using the second fundamental theorem of calculus, which states that if   f is continuous on   [  a  ,  b  ] and   a  ≤  x  ≤  b the derivative of an integral of   f can be calculated       d          d      x            ∫          a              x        f  (  t  )  d  t  =  f  (  x  ):  cos  ⁡      u    ×            d      u              d      x      Rewrite the derivative:  cos  ⁡      u    ×      d          d      x        (  u  )Substitute back   u  =      x    3   We can do this because the substituted values don’t represent the actual solution (they just simplify the calculation of one part of the problem!).   cos  ⁡            x      3        ×      d          d      x        (      x    3    )Find the derivative of       x    3   by using the differentiation rule      d          d      x            x    n    =  n      x          n      −      1        cos  ⁡            x      3        ×  3      x    2  Use the commutative property to reorder the terms:  3      x    2    ×  cos  ⁡            x      3      The derivative of an integral       d          d      x            ∫                  π        2                            x        3              cos  ⁡  (  t  )  d  t is:  3      x    2    ×  cos  ⁡            x      3

Find the derivative of an integral: (d)/(dx) int_ (pi)/(2))^(x^3) cos(t)dt

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