13 . Leibnitz Rule for Differentiation Under Integral Sign
( a ) If Φ ( x ) and ψ ( x ) are defined on [ a , b ] and differentiable for every x and f ( t ) is continuous , then
( b ) If Φ ( x ) and ψ ( x ) are defined on [ a , b ] and differentiable for every x and f ( t ) is continuous , then
14 . If f ( x ) ≥ 0 on the interval [ a , b ], then 15 . If ( x ) ≤ Φ ( x ) for x ∈ [ a , b ], then
16 . If at every point x of an interval [ a , b ] the inequalities
g ( x ) ≤ f ( x ) ≤ h ( x ) are fulfilled then ,
18 . If m is the least value and M is the greatest value of the function f ( x ) on the interval [ a , bl . ( estimation of an integral ), then
19 . If f is continuous on [ a , b ], then there exists a number c in [ a , b ] at which
is called the mean value of the function f ( x ) on the interval [ a , b ].