intergal properities (square root)

Question

I have this question below ,I don't know if this thing right or not but i've just wonders about it If we have $$ A=\int \sqrt(f_x) $$ Can we do this move $$ A^2=\int f_x $$ Thanks

Answer 1

No

Counterexample:

$\int xdx = \frac{1}{2}x^2$
$\int x^2dx = \frac{1}{3}x^3$

$(\frac{1}{2}x^2)^2=\frac{1}{4}x^4\neq\frac{1}{3}x^3$

Answer 2

No. As an analogy, we can consider A as a simple summation. $$A = x_1 + x_2 + ... + x_n$$ $$A^2 = (x_1 + x_2 + ... + x_n)^2\neq x_1^2 + x_2^2 + ... + x_n^2$$