If $f,g$ is Riemann integrable on [a,b], then also $\frac{|f(x)|}{1+f(x)^2+g(x)^2}$ is also Riemann integrable on [a,b]

Question

I'm homelearning calculus and trying to prove or disprove the following statement:

If $f,g$ is Riemann integrable on [a,b], then also $\frac{|f(x)|}{1+f(x)^2+g(x)^2}$ is also Riemann integrable on [a,b].

I'm really not sure how to approach this, could you please help me?

Thanks

Answer 1

Hint: We need to use 4 facts:

• if $f$ is continuous and $g$ is Riemann-integrable then $f \circ g$ is Riemann integrable
• if $f$ and $g$ are Riemann-integrable then $fg$ is Riemann-integrable
• if $f$ is Riemann integrable and $\exists \delta > 0$ so that $|f| > \delta$ then $1/f$ is Riemann-integrable
• if $f$ and $g$ are Riemann-integrable then so is their linear combination

(Note: the third is a consequence of the first one, but people like to mention it as a separate one)