Proving some basic facts about integrals Let $f$, $g$ be Riemann integrable functions on the interval $[a,b]$, that is $f,g \in \mathscr{R}([a,b])$.
(i) $\int_{a}^{b} (cf+g)^2\geq 0$ for all $c \in \mathbb{R}$.
(ii)  $2|\int_{a}^{b}fg|\leq c \int_{a}^{b} f^2+\frac{1}{c}\int_{a}^{b} g^2$ for all $c \in \mathbb{R}^+$  
I don't have a complete answer for either of these, but I have some ideas.
For (i) $\int_{a}^{b} (cf+g)^2=\int_{a}^{b}c^2f^2+2cfg+g^2=c^2\int_{a}^{b}f^2+c\int_{a}^{b}2fg+\int_{a}^{b}g^2$.  The first third is positive because if $c <0$ then $c^2>0$.  I'm not sure about the middle third.  The final third is positive.
I did notice that if I can figure out (i)...(ii) follows from some rearranging.

I will assume that the functions are real-valued.
For the first point, I will use that if an R-integrable function $h$ is nonegative on $[a,b]$, then $\int_a^bh(x)dx\geq 0$.
For the second point, I will use that if $h(x)\leq k(x)$ on $[a,b]$, then $\int_a^bh(x)dx\leq \int_a^bk(x)dx$. Note that the latter follows readily from the former by linearity of the integral.
1) We have $(cf(x)+g(x))^2\geq 0$ for all $x\in [a,b]$, so $\int_a^b(cf(x)+g(x))^2dx\geq 0$.
2) Recall that $2|ab|\leq a^2+b^2$ for every $a,b\in\mathbb{R}$.
With $a=\sqrt{c}f(x)$ and $b=g(x)/\sqrt{c}$ ,this yields
$$
2|f(x)g(x)|\leq cf(x)^2+\frac{1}{c}g(x)^2
$$
on $[a,b]$.
Hence
$$
2\int_a^b|f(x)g(x)|dx\leq c\int_a^bf(x)^2dx+\frac{1}{c}\int_a^bg(x)^2dx.
$$
Finally, we have $|\int_a^bf(x)g(x)dx|\leq \int_a^b|f(x)g(x)|dx$, hence the second inequality.
Note: As you said, you can also deduce 2) from 1) directly by expanding $(cf+g)^2$ and then dividing by $c$. But $2|ab|\leq a^2+b^2$ is so useful I could not resist mentioning it.
