If functions f(x) and g(x) are continuous on the interval [a, b] and g(x) retain the same sign on [a, b] then there is c $€$(a,b) such that ${\int }_a^b g\left(x\right)f\left(x\right)dx=f\left(c\right){\int }_a^b g\left(x\right)dx$ This is known as Mean-Value theorem. This result can be used to estimate some definite integrals. Other results which can be used for estimation are ; 
i) If f increases and has a concave graph in the interval [a, b] then  $(\mathrm{b}-\mathrm{a})\mathrm{f}\left(\mathrm{a}\right)<{\int }_{\mathrm{a}}^{\mathrm{b}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}<(\mathrm{b}-\mathrm{a})\frac{\mathrm{f}\left(\mathrm{a}\right)+\mathrm{f}\left(\mathrm{b}\right)}{2}$ 
ii) If f increases and has a convex graph in the interval [a, b] then $(\mathrm{b}-\mathrm{a})\frac{\mathrm{f}\left(\mathrm{a}\right)+\mathrm{f}\left(\mathrm{b}\right)}{2}<{\int }_{\mathrm{a}}^{\mathrm{b}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}<(\mathrm{b}-\mathrm{a})\mathrm{f}\left(\mathrm{b}\right)$
iii) ${\left|{\int }_{\mathrm{a}}^{\mathrm{b}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{g}\left(\mathrm{x}\right)\mathrm{dx}\right|}^2\le \left({\int }_{\mathrm{a}}^{\mathrm{b}} {\mathrm{f}}^2\left(\mathrm{x}\right)\mathrm{dx}\right)\left({\int }_{\mathrm{a}}^{\mathrm{b}} {\mathrm{g}}^2\left(\mathrm{x}\right)\mathrm{dx}\right)$