Find the derivative of ∫ 3 x sin 3 ⁡ t d t

Question

Find the derivative of       ∫    3    x              sin      3        ⁡    t    d  t

pulpasqsltl · Accepted Answer

We want to evaluate      d          d      x                  [            ∫    3    x        sin    3    ⁡  (  t  )  d  t            ]      By the fundamental theorem of calculus, we can &quot;replace&quot; the   t and   x, to arrive at the answer      sin    3    ⁡  (  x  )However, if the question was to simply evaluate the integral      ∫    3    x        sin    3    ⁡  (  t  )  d  tthen we have                              cos          3                ⁡                  (          x          )                −        3        cos        ⁡                  (          x          )                    3        −                              cos          3                ⁡                  (          3          )                −        3        cos        ⁡                  (          3          )                    3      The two procedures above are very different, since taking the derivative of an integral (i.e. applying       d          d      x      ) is to simply &quot;undo&quot; the integrating process and be left with the original function, whereas integrating the function is to find the antiderivative.

Matthew Hubbard · Accepted Answer

The Fundamental Theorem of Calculus states that if

g (x) = \int_{a}^{f (x)} h (t) d t

where

a

is any constant, then

g^{'} (x) = h (f (x)) \cdot f^{'} (x)

Using this with the integral,

g (x) = g (x)

,

f (x) = x

, and

h (x) = \sin^{3} t

. We may now plug in these values.

g^{'} (x) = \sin^{3} (x)

So,

\frac{d}{d x} [\int_{3}^{x} \sin^{3} (t) d t] = \sin^{3} (x)

Find the derivative of <msubsup> ∫ 3 x </msubsup> <mrow cl

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