Answer
FALSE
Work Step by Step
Given:$\int_{a}^{b}[f(x)g(x)]dx=(\int_{a}^{b}f(x))dx)(\int_{a}^{b}g(x)dx$
Consider $f(x)=x,g(x)=x^{2}, a=0, b=1$
Take Left Side:
$\int_{0}^{1}(x)(x^{2} )dx=\int_{0}^{1}(x^{3} )dx$
$=[\frac{x^{4}}{4}]_{0}^{1}$
$=\frac{1}{4}$
Take right side:
$(\int_{0}^{1}(x))dx)(\int_{0}^{1}(x^{2})dx=[\frac{x^{2}}{2}]_{0}^{1}[\frac{x^{3}}{3}]_{0}^{1}$
$=\frac{1}{2} \times \frac{1}{3}$
$=\frac{1}{6}$
Thus, $\frac{1}{4}\ne\frac{1}{6}$
Hence, $\int_{a}^{b}[f(x)g(x)]dx\ne(\int_{a}^{b}f(x))dx)(\int_{a}^{b}g(x)dx$
The given statement is false.