Antiderivatives and definite integrals - 2Prove that the function f ( x ) = x 2 ∫ 0 1 t sin 2 ⁡ ( t x ) d tis differentiable in R and determine the formula f′(x) of its derivative.

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Antiderivatives and definite integrals - 2Prove that the function   f  (  x  )  =      x    2        ∫          0              1        t      sin    2    ⁡  (  t  x  )  d  tis differentiable in       R   and determine the formula f′(x) of its derivative.

Strehaiau10p7 · Accepted Answer

Step 1For all   x  ∈      R   we have  f  (  x  )  =      x    2        ∫          0              1        t      sin    2    ⁡  (  t  x  )  d  t  =      ∫          0              1            x    2    t      sin    2    ⁡  (  t  x  )  d  t  =      ∫          0              1        (  x  t  )      sin    2    ⁡  (  t  x  )  x  d  tStep 2Now, let   u  =  x  t thus   d  u  =  x  d  t (x is considered to be a constant when integrating with respect to t). For   t  =  0  ,  1, we get   u  =  0  ,  x, respectively. Hence   f  (  x  )  =      ∫          0              x        u      sin    2    ⁡  u  d  u  ,                          x  ∈      R  Consequently, f is a differentiable function in       R   and its derivative is given by       f    ′    (  x  )  =  x      sin    2    ⁡  x.

Antiderivatives and definite integrals - 2 Prove that the function f ( x ) =

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