偶倍奇零例3. 设 f(x)EC[-a,a],(1)若f(-x)= f(x), 则[~ f(x)dx=2[f(x)dx(2)若f(-x)=-f(x),则(~ f(x)dx=0: {~ f(x)dx= ~ f(x)dx + f°f(x)dx证:令x=-t= °f(-t)dt + J°f(x)dxJtf(-x)+ f(x)]dx2J°f(x)dx， f(-x)=f(x)时0.f(-x)=-f(x)时o0ol0l0x机动自录上页下页返回结束
例3. 证: (1) 若 − = a a a f x x f x x 0 则 ( )d 2 ( )d = − f x x a a ( )d (2) 若 ( )d = 0 − a a 则 f x x f x x a ( )d 0 − f x x a ( )d 0 + f t t a ( )d 0 = − f x x a ( )d 0 + f x f x x a [ ( ) ( )]d 0 = − + f (−x) = f (x)时 f (−x) = − f (x)时 偶倍奇零 机动 目录 上页 下页 返回 结束 令x = −t =
二、定积分的分部积分法定理2. 设u(x), v(x) ε C'[a, b],则bbu'(x)v(x)dxu(x)v(x)dx = u(x)v(x)aa证: : [u(x)v(x)}' = u'(x)v(x)+u(x)v'(x)两端在[a,b]上积分.bu'(x)v(x)dx +u(x)v'(x)dxAqa66u(x)v'(x)dx = u(x)v(x)u'(x)v(x)dxaoeol00x机动自录上页下页返回结束
二、定积分的分部积分法 定理2. ( ), ( ) [ , ], 1 设u x v x C a b 则 a b 证: [u(x)v(x)] = u (x)v(x) + u(x)v (x) u(x)v(x) a b u x v x x u x v x x b a b a = ( ) ( )d + ( ) ( )d = u(x)v(x) a b − b a u (x) v(x)dx 机动 目录 上页 下页 返回 结束 两端在[a,b]上积分