? E?cc?M ￥s x1c?M ￥?ès 1. p/ K (1) lima!0R1?1px2 +a2dx (2) lima!0R20 x2 cosax dx (3) lima!0R1+aa dx1+x2+a2. 2 pF0(x) ? (1) F(x) = Rx2x e?xy2dy (2) F(x) = Rcosxsinx ex p 1?y2dy (3) F(x) = Rb+xa+x sin(xy)y dy (4) Rx0 hRx2 t2 f(t;s)ds i dt. 3 !f(x)1 ??f ? F(x) = 1h2 Z x 0 ?Z x 0 f(x+? +·)d· ? d? pF00(x). 4ù?f ? F(y) = Z 1 0 yf(x) x2 +y2dx ￥ ??? ?f(x) ^[01]  ?? O1?￥f ?. 5?¨s|/ p?E p/ s (1) R … 20 ln(a2 ?sin2 x)dx (a > 1); 1 (2) R…0 ln(1?2acosx+a2)dx(jaj < 1) (3) R … 20 ln(a2 sin2 x+b2 cos2 x)dx (a;b 6= 0) (4) R … 20 arctan(atanx) tanx dx(jaj < 1). 6?¨s?DQ? p/ s (1) R10 xb?xalnx dx (a > 0;b > 0) (2) R10 sin?ln 1x￠ xb?xalnx dx (a > 0;b > 0). 7 !f1 V±f ? k p/ f ?￥=¨? ? (1) F(x) = Rx0 (x+y)f(y)dy (2) F(x) = Rba f(y)jx?yjdy (a < b) 8￡ üR10 dxR10 x2?y2(x2+y2)2dy 6= R10 dyR10 x2?y2(x2+y2)2dx. 9 !F(y) = R10 lnpx2 +y2dxù ^?? ? F0(0) = Z 1 0 @ @y ln p x2 +y2jy=0dx: 10 ! F(x) = Z 2… 0 excos cos(xsin )d p￡F(x) · 2…. 11 !f(x)1 Q V±f ?’(x)1 V±f ?￡ üf ? u(x;t) = 12[f(x?at)+f(x+at)]+ 12a Z x+at x?at ’(z)dz ?@???Z? @2u @t2 = a 2@2u @x2 # SHq u(x;0) = f(x);ut(x;0) = ’(x): 2 x2ccc???MMM   ￥￥￥<<<lllsss 1￡ ü/ s·?￥ uW =Bá l ? (1) R+10 cos(xy)x2+y2 dy (x ? a > 0) (2) R+10 cos(xy)1+y2 dy (?1 < x < +1) (3) R+11 yxe?ydy (a ? x ? b) (4) R+11 e?xycosyyp dy (p > 0;x ? 0) (5) R+10 sinx21+xpdx (p ? 0). 2) ?/ s·? uW ￥Bá l ?? (1) R+10 pfie?fix2dx (0 < fi < +1) (2) R+10 xe?xydy  ix 2 [a;b] (a > 0) iix 2 [0;b] (3) R+1?1 e?(x?fi)2dx  ia < fi < b ii?1 < fi < +1 (4) R+10 e?x2(1+y2) sinxdy (0 < x < +1). 3 !f(t)t > 0 ??R+10 t?f(t)dt?? = a;? = b￥ l ? Oa < bb p￡R+10 t?f(t)dt1??[a;b]Bá l ?. 4) ?/ f ?·? uW ￥ ??? (1) F(x) = R+10 xx2+y2dyx 2 (?1;+1) 3 (2) F(x) = R+10 y21+yxdyx > 3 (3) F(x) = R…0 sinyyx(…?y)2?xdyx 2 (0;2). 5 ?f(x;y)[a;b]￡[c;+1)  ??c?M <ls I(x) = Z +1 c f(x;y)dy [a;b) l ?x = b H? ?￡ üI(x)[a;b)?Bá l ?. 6c?M ￥<lsI(x) = R+1c f(x;y)dy[a;b]Bá l ?￥ 1H q ^ ?B t?+1￥?9 ? fAng ?A1 = cf ?[) ? 1X n=1 Z An+1 An f(x;y)dy = 1X n=1 un(x) [a;b] Bá l ?. 7¨ 5￥2 ?￡ üc?M <lsI(x) = R+1c f(x;y)dy[a;b]￥ s?DQ?? ? ? ?19.12?s|/ p? ?? ? ? ?19.13. 8 ?¨±s?DQ?9 ?/ s (1) In(a) = R+10 dx(x2+a)n+1 n1?? ?a > 0 (2) R+10 e?ax?e?bxx sinmxdx a > 0;b > 0 (3) R+10 xe?fix2 sinbxdx (fi > 0). 9¨? ?￥sE9 ?/ s (1) R+10 e?ax 2?e?bx2 x dx a > 0;b > 0 (2) R+10 e?ax?e?bxx sinmxdx a > 0;b > 0. 10 ?¨11+x2 = R+10 e?y(1+x2)dy9 ? ? ? ? ?s L = Z +1 0 cosfix 1+x2dx 4 ? L1 = Z +1 0 xsinfix 1+x2 dx: 11 ?¨1px = 2p… R+10 e?xy2dy(x > 0)9 ?° ~ \:s F = Z +1 0 sinx2dx = 12 Z +1 0 sinxp x dx ? F1 = Z +1 0 cosx2dx = 12 Z +1 0 cosxp x dx: 12 ?¨X?s Z +1 0 sinx x dx = … 2 Z +1 0 e?x2dx = p… 2 9 ?/ s (1) R+10 sin4 xx2 dx (2) 2… R+10 sinycosyxy dy (3) R+10 x2e?fix2dx (a > 0) (4) R+10 e?(ax2+bx+c)dx (a > 0) (5) R+1?1 e?(x2+a 2 x2)dx (a > 0). 13 p/ s (1) R+10 1?ett costdt (2) R+10 ln(1+x2)1+x2 dx. 14￡ ü (1) R10 ln(xy)dy[1b;b] (b > 1) Bá l ? (2) R10 dxxy(?1;b] (b < 1) Bá l ?. 5 x3 x x x ? ? ?sss 1 ?¨ x ?s9 ?/ s (1) R10 dxp 1?x14  (2) R10 px?x2dx (3) R10 px3(1?px)dx; (4) Ra0 x2pa2 ?x2)dx (a > 0) (5) R … 20 sin6 xcos4 xdx (6) R+10 dx1+x4 (7) R+10 x2ne?x2dx n1?? ? (8) R…0 dxp3?cosx (9) R … 20 sin2n xdx n1?? ? (10) R10 xm?ln 1x￠n?1 dx (n1?? ?). 2|/ s¨ x ?sV Ui ps￥i× (1) R+10 xm?12+xndx (2) R10 dxnp1?xm (3) R … 20 tann xdx (4) R10 ?ln 1x￠p dx (5) R+10 xpe?fix lnxdx (fi > 0). 3￡ ü 6 (1) R+1?1 e?xndx = 1n?(1n) (n > 0) (2) limn!+1R+1?1 e?xndx = 1. 4￡ ü B(a;b) = Z 1 0 xfi?1 +xb?1 (1+x)a+b dx ?(fi) = sfi Z +1 0 xfi?1e?sxdx (s > 0): 7