Integral proof of an exponent property.

Question

Let function $f$ be continuous everywhere. Prove or disprove (by finding a counter example) that: $$ \ \int (f(x))^2 \, dx = \left(\int f(x) \, dx\right)^2 $$

I'm not really sure how to go about this problem, would it require a Darboux / Riemann proof? Anything helps. Thanks.

Answer 1

Let $f(x)=x$. Them LHS $=\frac {x^{3}} 3+C$ and RHS $=(\frac {x^{2}} 2+C)^{2}$. Are these equal ?

Answer 2

You can try $f(x)=1$ to disprove the claim. It gives $$\int f^2(x)\,dx=\int dx=x+c$$ $$\left(\int f(x) \, dx\right)^2=(x+c)^2$$ The claim is false.