Calculate a, b, (real numbers) if the function F'= f f = (2x^2 + 1)/sqrt(1+x^2) F= (ax + b)*sqrt(1 + x^2)

To determine a and b, we'll differentiate
F(x).

We notice that F(x) is a product, so we'll
differentiate using the product rule.

(u*v) = u'*v +
u*v'

u = ax + b => u' =
a

v = sqrt(1 + x^2)

Since v is
a composed function, we'll use the chain rule to calculate it's
derivative.

v' =
[1/2sqrt(1+x^2)]*(1+x^2)'

v' =
2x/2sqrt(1+x^2)

We'll
simplify:

v' =
x/sqrt(1+x^2)

F'(x) = a*sqrt(1 + x^2) + (ax +
b)*x/sqrt(1+x^2)

F'(x) = [a(1+x^2) +
x(ax+b)]/sqrt(1+x^2)

We know, from enunciation,
that:

F'(x) = f(x)

f = (2x^2 +
1)/sqrt(1+x^2)

[a(1+x^2) + x(ax+b)]/sqrt(1+x^2) = (2x^2 +
1)/sqrt(1+x^2)

We'll simplify like
terms:

a(1+x^2) + x(ax+b) = 2x^2 +
1

We'll remove the brackets from the left
side:

a + ax^2 + ax^2 + bx = 2x^2 +
1

a + 2ax^2 + bx = 2x^2 +
1

The expression from the left side is equal to the
expression from the right side, if and only if the coefficients of the correspondent
terms are equal.

2a = 2

We'll
divide by 2:

a = 1

b*x =
0*x

b = 0

So,
F(x) = x*sqrt(1 + x^2), so that F'(x) = f(x).