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when F(x)=ʃ(subscript 0)(superscript x) f(t)dt. find F(x).

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inovermyhead | Student, Undergraduate | (Level 1) eNoter

Posted August 9, 2012 at 7:16 PM via web

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when F(x)=ʃ(subscript 0)(superscript x) f(t)dt. find F(x).

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted July 24, 2013 at 6:06 PM (Answer #1)

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You should use the fundamental theorem of calculus to evaluate the definite integral, such that:

`int_0^x f(t)dt = F(t)|_0^x => int_0^x f(t)dt = F(x) - F(0)`

Since the problem does not specify the equation of the function `f(t)` , allow me to select the function `f(t) = sqrt t` , whose domain of definition is `[0,oo)` , such that:

`int_0^x f(t)dt = int_0^x sqrt t dt `

You should evaluate the definite integral, such that:

`int_0^x sqrt t dt = int_0^x t^(1/2) dt = (t^(1/2+1))/(1/2+1)|_0^x`

`int_0^x sqrt t dt = (t^(3/2))/(3/2)|_0^x`

`int_0^x sqrt t dt = (2/3)(x^(3/2) - 0)`

`int_0^x sqrt t dt = (2xsqrtx)/3`

Hence, evaluating the function `F(x)` , using the considered function `f(t) = sqrt t, t >= 0` and the fundamental theorem of calculus, yields `int_0^x sqrt t dt = (2xsqrtx)/3` .

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