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1、 A (1) 1 21 1/29 1 20 ?: ?5 (Y) e f Ra,b, K |f| Ra,b ? ? ? Z b a f(x)dx ? ? ? 6 Z b a |f(x)|dx. e f,g Ra,b, K fg Ra,b. (Cauchy ?) e f,g Ra,b, K ?Zb a f(x)g(x)dx ?2 6 ?Zb a (f(x)2dx ?Zb a (g(x)2dx ? . 2/29 (H older) ef,gCa,b, p,q1, 1 p+ 1 q = 1, K ? ? ? Z b a f(x)g(x)dx ? ? ? 6 ?Zb a |f(x)|pdx ?1 p?
2、Z b a |g(x)|qdx ?1 q . (r?1n) e f Ra,b 3 (a,b) SY, K (a,b) ? Z b a f(x)dx = f()(b a). (21n) XJf,g Ca,b g C, K a,b ? Z b a f(x)g(x)dx = f() Z b a g(x)dx. 3/29 ?: ?n ? I m, ? F,f : I R . e F 3 I Y, 3 I ?S? F 0 = f, K F f ?. ? f Ra,b. x a,b, -F(x) =R x a f(t)dt, K F Ca,b. XJf 3: x0 a,b Y, K F 3: x0? F
3、0(x0) = f(x0). e f 3: x0=kY, K F 3T:k A?. 3a?m:?Xd. 4/29 e f Ca,b, K F C (1)a,b F0 = f, = F f 3 a,b ?. ? f Ca,b, ? , : , a,b ?. u , G(u) = R (u) (u) f(t)dt. K G ? u , k G0(u) = f ?(u)?0(u) f?(u)?0(u). ;.f: lim x0 1 x R x 0 sin3t t dt = 3. 5/29 (Newton-Leibniz ) b? f Ca,b, ? G Ca,b f ?, K R b a f(x)d
4、x = G? ?b a := G(b) G(a). 3m? f ? L f ?, P R f(x)dx. x gC?. e f Ca,b, K R f(x)dx = R x a f(t)dt + C. 6/29 ?!?X e R f(x)dx = F(x) + C, K F 0(x) = f(x), ?Z f(x)dx ?0 = F 0(x) = f(x), dF(x) = f(x)dx, d ?Z f(x)dx ? = f(x)dx, Z f(x)dx = Z F 0(x)dx = Z dF(x) = F(x) + C. (55) , R, k Z (f(x) + g(x)dx = Z f(
5、x)dx + Z g(x)dx. 7/29 ? R dx = x + C. R xdx = x+1 +1 + C ( 6= 1), R 1 x dx = log|x| + C. R axdx = ax loga + C (a 0, a 6= 1), R exdx = ex+ C. R sinxdx = cosx + C, R cosxdx = sinx + C. 8/29 R shxdx = chx + C, R chxdx = shx + C. R sec2xdx = tanx + C. R csc2xdx = cotx + C. R dx 1x2= arcsinx + C. R dx 1+
6、x2 = arctanx + C. R dx x2+a2= log|x + x2 + a2| + C. R dx x2a2= log|x + x2 a2| + C. 9/29 K 1. ka?m:?vk?. y: y, b? f k?3 :x0a?. ”5,?f(x0+0)f(x00), K f. ? ?f(x 0 0),f(x0+ 0) ? ? 6= f(x0). f(x0 0) . q 6= f(x0), u f 3 c,d ? , Darboux ng! ?y(. 10/29 1. 1? (n): e F 0(y) = f(y), ? u ?, K (F u)0(x) = F 0(u(x
7、)u0(x) = f(u(x)u0(x). ? R f(u(x)u0(x)dx = F(u(x) + C. ? R f(u(x)du(x), Kk Z f(u(x)u0(x)dx = Z f(u(x)du(x) = F(u(x) + C. 11/29 8. O R 2xex 2dx. ): R 2xex 2dx =R ex 2d(x2) =R d(ex 2) = ex2 + C. 9. ? a 6= 0. O R dx a2+x2. ): R dx a2+x2 = 1 a R d(x a) 1+(x a)2 = 1 a R d?arctan x a ? = 1 a arctan x a + C
8、. 10. ? a 0. O R dx a2x2. ): R dx a2x2= R d(x a) 1(x a) 2 = Rd(arcsin x a) = arcsin x a + C. 12/29 11. O R tanxdx. ): R tanxdx = R sinx cosxdx = R d(cosx) cosx = R d?log|cosx|?= log|cosx| + C. 12. O R cotxdx. ): R cotxdx=R cosxdx sinx =R d(sinx) sinx = log|sinx|+C. 13. O R tan2xdx. ): R tan2xdx = R
9、(sec2x 1)dx = R d(tanx x) = tanx x + C. 13/29 14. O R dx (1+4x2)(arctan2x+1)2. ): R dx (1+4x2)(1+arctan2x)2 = 1 2 R d(2x) (1+(2x)2)(1+arctan2x)2 = 1 2 R darctan2x (1+arctan2x)2 = 1 2 R d(1+arctan2x) (1+arctan2x)2 = 1 2 R d? 1 1+arctan2x ? = 1 2(1+arctan2x) + C. 14/29 15. O R dx sinx. ): 1. R dx sinx
10、 = R dx 2sin x 2 cos x 2 = R d(x 2) tan x 2 cos2 x 2 = R d(tan x 2) tan x 2 = log|tan x 2| + C. 2. R dx sinx = R sinxdx sin2x = R d(cosx) 1cos2x = 1 2 R? 1 cosx1 1 cosx+1 ?d(cosx) = 1 2 log ? ?cosx1 cosx+1 ? ?+C = 1 2 log ? ?(cosx1)2 cos2x1 ? ? + C = log ? ?1cosx sinx ? ? + C = log|cscx cotx| + C. 1
11、5/29 16. O R x 3+2xx2dx. ): R x 3+2xx2dx = R ? 1 2 (3+2xx2)0 3+2xx2 + 1 3+2xx2 ?dx = 1 2 R d(3+2xx2) 3+2xx2 + R 1 4(x1)2 ?dx = 3 + 2x x2+ R 1 1(x1 2 )2 d(x1 2 ) = 3 + 2x x2+ arcsin x1 2 + C. 17. O R dx 1+ex. ): R dx 1+ex = R d(ex) ex+1 = log(1 + ex) + C. 16/29 18. O R secxdx. ): R secxdx = R dx cosx
12、 = R d(x+ 2) sin(x+ 2) = log|csc(x + 2) cot(x + 2)| + C = log|secx + tanx| + C. K: 1 156 1 2 K 1 (1), (2) ?K, 1 3 K1 (1), (6), (7), (9) ?K, 1 4 K1 (3), (4), (9), (11) ?K, O R secxdx. 17/29 2. 1?: R f(x)dx x=x(t) = R f(x(t)x0(t)dt = F(t) + C t=t(x) =F(t(x) + C. 19. O R sin x x dx. ): R sin x x dx t=x
13、 = R sint t d(t2) = R sint t (2t)dt = R 2sintdt = 2cost + C = 2cos x + C. 18/29 20. O R dx 1+ 3 1+x. ): R dx 1+ 3 1+x u= 3 1+x = R d(u31) 1+u = R 3u2du 1+u = 3 R ?u 1 + 1 1+u ?du = 3?1 2u 2 u + log|1 + u|?+ C = 3?1 2(1+x) 2 3 3 1 + x+log|1+ 3 1 + x|? +C. 19/29 21. O R x2 1x+1dx. ): R x2 1x+1dx u=x+1
14、 = R u23 1u d(u2 1) = R ? 2u2 2u + 4 4 1u ?du = 2 3u 3 u2+ 4u + 4log|1 u| + C1 = 2 3(x5) x + 1x+4log|11 + x|+C. 5: ?k a2 x2, x2 a2, n?“. 20/29 22. O R a2 x2dx (a 0). ): R a2 x2dx x=asint = |t| 6 2 R p a2 a2sin2td(asint) = R (acost) (acost)dt = a2 R cos2tdt = a2 R 1+cos2t 2 dt = a2 2 ?t + sin2t 2 ? + C = a2 2 ?t + sintcost? + C = a2 2 arcsin x a + x 2 a2 x2+ C. 21/29 23. O R dx x2+a2 (a 0). ): R dx x2+a2 x=atant = |t| 0). ): ? (,a)(a,+). ? x a ,
