(" Vsin' x - sin' xdx例2变形f" cos x(sin x) dx去绝对值F2 cos x(sinx) dx-f" cos x(sin x)zdx=2分f (sinx)id sinx -JJ" (sin x)id sin x22-(sin x)sin x)2=55元26/25
例2 − 0 3 5 sin x sin xdx ( ) = 0 2 3 cos x sin x dx 变 形 去绝对值 = ( ) − 2 2 3 cos x sin x dx ( ) = 2 0 2 3 sin sin x d x 凑 分 ( ) − 2 2 3 sin x d sin x ( ) 2 0 2 5 sin 5 2 = x ( ) − 2 2 5 sin 5 2 x . 5 4 = ( ) 2 0 2 3 cos sin x x dx 6/25
3dxe例3ex /lnx(1- Inx)3凑分re4d(ln x)=/in x(1 - In x)33d(lnx)d /In x= 2JJreJnx (1-Inx)/1-(/lnx)3T=2[aresin(/Inx)] = 7/25
例3 − 4 3 ln (1 ln ) e e x x x dx 凑分 = − = 4 3 ln (1 ln ) e (ln ) e x x d x − = 4 3 2 1 ( ln ) ln 2 e e x d x 4 3 2 arcsin( ln ) e e = x . 6 = − 4 3 ln (1 ln ) e (ln ) e x x d x 7/25
1V例4dx(a > 0)J022-xx+va去根式元acost12dt=x=asint,te[0,]Joasint + a'(l-sin't)元-2元cost(sint +cost)+(cost -sint)2dt =dt12 Josint+ costsint +cost11cost-sint元儿[nsint + cos t]-= "dt=222sint + cost8/25
例4 + − a dx a x a x 0 2 2 ( 0) 1 去根式 sin , [0, ] 2 = = x a t t + = 2 0 sin cos cos dt t t t + + + − = 2 0 sin cos (sin cos ) (cos sin ) 2 1 dt t t t t t t 2 0 lnsin cos 2 1 2 2 1 + + = t t . 4 = + − 2 0 2 2 sin (1 sin ) cos dt a t a t a t + − = + 2 0 sin cos cos sin 1 2 1 dt t t t t 8/25
例5证明：设f(x)在[-a,al上连续，①若f(x)为偶函数，则"f(x)dx =2J"f(x)dx;②若f(x)为奇函数，则["f(x)dx=0.证f" f(x)dx = f" f(x)dx + f" f(x)dx,=" T° f(-t)(-dt)+ J" f(x)dx= J f(-x)dx +J° f(x)dx= J"(F(-x)+ F(x)dx=[2, () 个图数证毕。0，f(x)为奇函数。9/25
例 5 证明：设 f (x)在[−a, a]上连续， ①若 f (x)为偶函数，则− = aa a f x dx f x dx 0 ( ) 2 ( ) ； ②若 f (x)为奇函数，则− = aa f (x)dx 0. 证 ( ) ( ) ( ) , 0 0 − − = + aa a a f x dx f x dx f x dx = − − =− 0 ( )( ) a x t f t dt + a f x dx 0 ( ) = − + a f x dx 0 ( ) a f x dx 0 ( ) = − + a f x f x dx 0 ( ( ) ( )) ={2 0 f (x)dx, f (x)为偶函数； a 0 , f (x)为奇函数。 证毕。 9/25