# Consider the functions: F(x)= definite integral [1,x] of f(t)dt f(t)= definite integral [1,t^2] of (sqrt(5+u^4)/u) du Find F''(1). Use the Fundamental Theorem of Calculus

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You need to evaluate f(t), hence, you need to solve the definite integral int_1^(t^2)(sqrt(5+u^4))/u du  such that:

u^4 =v => 4u^3du = dv => du = (dv)/(4u^3)

int (sqrt(5+u^4))/u du = int (sqrt(5+v))/u(dv)/(4u^3)

int (sqrt(5+v))/(4u^4) dv = (1/4)int (sqrt(5+v))/v(dv)

You should consider the next substitution such that:

sqrt(5+v) = s => s^2 = 5+v => v = s^2-5

1/(2sqrt(5+v))dv = ds => dv = 2sqrt(5+v)ds => dv = 2s ds

(1/4)int (sqrt(5+v))/v(dv) = (1/2)int s^2/(s^2-5)ds

You need to subtract and add 5 to numerator such that:

int (s^2-5+5)/(s^2-5)ds

Using the property of linearity yields:

int (s^2-5+5)/(s^2-5)ds = int (s^2-5)/(s^2-5)ds + int (5)/(s^2-5)ds

int (s^2-5+5)/(s^2-5)ds = int ds + 5 ln|(s-5)/(s+5)| + c

`(1/2)int s^2/(s^2-5)ds = s/2 +...

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