Review a a b a b a b a ?( b a ) f ( x )dx kf ( x )dx ( f ( x ) g ( x ))dx c f ( x )dx f ( x )dx b b a f ( x )dx ??( b a ) The fundamental Theorem of Calculus dF d x f ( x )dx ??? dx dx a d d x F ( x) f ( x )dx f ( x ) dx dx a x a f ( x )dx x a F ( x )dx ?? The Mean Value Theorem for Definite Integrals Indefinite integrals and the substitution rules*** Substitution in Definite Integrals n1 d u n du u dx n 1 dx n1 du u n u dx dx n 1 C Example 1 2 3 3 x x 1 dx 4t 1dt f ( g( x )) g( x )dx Let u g ( x ), du g ( x )dx . Then f ( g( x )) g( x )dx f ( u)du F (u) C F ( g ( x )) C Example 3 cos(7 5)d 2 3 x sin( x )dx Exercise on P375 Q13, Q14 Example 4 2 3 x sin( x ) dx 1 cos 2 2 x dx 2 zdz 3 z2 1 Example 5 2 sin xdx 2 cos xdx Example 7 2 g ( x ) sin x over the Figure 5.24 shows the graph of interval [0, 2 ]. Find (a) the definite integral of g ( x ) over [0, 2 ], (b)the area between the graph of the function and the x-axis over [0, 2 ]. Homework on P375 Q20, Q22, Q27, Q28;