What is f(subscript 1)(1) if f(subscript n)(x) = integral (0 to x) t^n sqroot(t^2+1)dt?

sciencesolve | Teacher | (Level 3) Educator Emeritus

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The problem provides the following information with respect to equation of the function `f_(n)(x)` , such that:

`f_(n)(x) = int_0^x t^n sqrt(t^2 + 1)dt`

You need to evaluate `f_1(1)` , hence, you need to replace 1 for `n` and 1 for `x` , in equation provided, such that:

`f_1(1) = int_0^1 t^1 sqrt(t^2 + 1)dt`

You should come up with the following substitution, such that:

`t^2 + 1 = u => 2tdt = du => tdt = (du)/2`

Changing the limits of integration yields:

`t = 0 => u = 1`

`t = 1 => u = 2`

`f_1(1) = int_1^2 sqrt u*(du)/2 => f_1(1) = 1/2 int_1^2 u^(1/2) du`

`f_1(1) = 1/2*(u^(3/2))/(3/2)|_1^2`

`f_1(1) = 1/3*u*sqrt u|_1^2`

Using the fundamental theorem of calculus, yields:

`f_1(1) = 1/3(2*sqrt2 - 1*sqrt1)`

`f_1(1) = (2*sqrt2 - 1)/3`

Hence, evaluating the given definite integral, under the given conditions, yields `f_1(1) = (2*sqrt2 - 1)/3` .