The Cauchy–Schwarz inequality for integrals

1. Expanding,
$$0\le\int_a^b(f+\lambda g)^2=\lambda^2\int g^2+2\lambda\int fg+\int f^2\quad\text{for all }\lambda\in\mathbb{R}.$$
A quadratic $A\lambda^2+B\lambda+C$ that is never negative has non-positive discriminant, $B^2-4AC\le0$ (if $A=\int g^2=0$, then $g=0$ and the inequality is trivial). Here $B=2\int fg$, so
$$4\left(\int fg\right)^2-4\int g^2\int f^2\le0.$$

2. Take $f=\sqrt{\sin x}$ and $g=1$ on $[0,\frac\pi2]$:
$$\left(\int_0^{\pi/2}\sqrt{\sin x}\,dx\right)^2\le\int_0^{\pi/2}\sin x\,dx\cdot\int_0^{\pi/2}1\,dx=1\cdot\frac\pi2.$$
So the integral is at most $\sqrt{\pi/2}\approx1.2533$.

Remark. The true value is about $1.198$, so the bound is quite sharp. It is attained exactly when $f$ and $g$ are proportional.