Definite

\hspace{-16}$Using $\bf{\int_{a}^{b}\left\{f(x)+\lambda\cdot g(x)\right\}^2 \geq 0\;\;,}$ where $\bf{\lambda \in \mathbb{R}-\{0\}}$\\\\\\ $\bf{\Rightarrow \int_{a}^{b}f^2(x)dx+\lambda^2 \cdot \int_{a}^{b}g^2(x)dx+2\lambda\cdot \int_{a}^{b}f(x)\cdot g(x)dx\geq 0}$\\\\\\ Now Let $\bf{\int_{a}^{b}f^2(x)dx = A}$ and $\bf{\int_{a}^{b}g^2(x)dx = B}$\\\\\\ and $\bf{\int_{a}^{b}f(x)\cdot g(x)dx = C}$.\\\\\\ So Equation is $\bf{\lambda^2 \cdot B+2 \lambda \cdot C+A \geq 0}$\\\\\\\ So Discriminant of given equation is $\bf{4C^2-4 \cdot B\cdot A \leq 0 }$\\\\\\ $\bf{\Rightarrow C^2\leq A\cdot B\Rightarrow C\leq \sqrt{A\cdot B}}$\\\\\\ So $\boxed{\boxed{\bf{\int_{a}^{b}f(x)\cdot g(x)dx\leq \sqrt{\int_{a}^{b}f^2(x)dx\cdot \int_{a}^{b}g^2(x)dx}}}}$