積分公式大全0001

1、一)含有 ax b的積分 ( a1dx 1 ax b a ln ax b2345678910111213常用積分公式0)1(ax b) dx a( 1)(axb)1)x1dx 2 (ax b bln ax b ax21 13ax ba3 2axb) Cdx 3 (ax b)2 2b(ax b) b2ln ax b Cax bdx 1ln ax b C x(ax b) bdx 1 a x2(ax b) bx b2lnx(ax b)12dx 2 (lna2(axx b)2dx a13 (axaxax bbax b)2bln ax baxb2 b) Cax bdx2x(ax b) b(ax b)含有

2、 ax b 的積分12 lnb2ax bax bdx 2 (ax b)33ax ax bdx 2 2 (3ax 2b15ax2 ax bdx 2 3 (15a2x2 12abx 8b2 ) (ax b)3 C105a3) (ax b)3 Cx2dx 2 (ax 2b) ax b C ax b3a215161718192021四）14dxx ax bdxx2 ax bax bdxxax b2x含有 x2dx2x2 3 (3a2x2 4abx 8b2 ) ax 15a3ax b bb ax b b2 arctan ax b bbCC (bC (b0)0)ax bbx2 ax bax b2a 的積分

3、1x2 = arctana a adx(x22na)dx2x含有2axdxax2 ba2bdxx ax bdxx ax bdx x ax2 2 2 n12(n 1)a2(x2a2)n 12n 322(n 1)a2dx2 2 n1 (x a )b(axaxa1C2aln0）的積分1 arctan a x C bax bab(b 0)1 ln2 ab ax bC (b 0)232x dx 1ax2 b 2aln ax2 b242x2ax bdx x adxax2 b25dxx(ax2 b)ln2b2x2ax26dxx2(ax2 b)1bx27dxa ln3 2 2 ln x3(ax2 b) 2b2

4、28dx22 (ax2 b)22930六）313233dx2ax2ax b2x22b(ax2 b)12b2bx2 Cdxax2 b含有 ax2bxdxax2 bx c2 x dx ax bx c含有 x2dxx2 a2dx2x x2c (a 0) 的積分a2 (a23a2)31 ln ax2aarshxadxadx22axbb24ac2axbb24acCbb2 4ac2arctan 2ax bC4ac b2 4ac b2(b2 4ac)2(b2 4ac)2 bx c0）的積分dx2a ax2 bx cC1 ln(xx2 a2 ) C2 x2 2 C2 2 2 axax2a2 C21 2 C22

5、 xa2x2x a2 dx x2 x22x dx 2 2 3 dx (x a )dx1lnax22xadxx2a2dx (x2a2)3dxx x2a2dx 2 2 2 x x a2a ln( xx2 a2)ln( xx2 a2 )x2 a2 a C22xa2axx (2x81(3x2x2 x2 a2dx 8(2x22xadxx22xa2 dx x含有 x2dxdxx2 a22ln( x x25a2) x223a ) Cx2 a2a2)34a8ln( xx2 a2 ) C22a4aln( x8x2 a2 ) Cx2 a2 a aln x xaaln( xx2 a2 )xxarchC1 =lnx

6、x2 a2xaCa2 (a 0) 的積分(x2 a2)3x a2 x2xx2 a2 dx35363738394041424344（七）4546474849505152535455565758(八)596022x x 2 2 a 2 2 dx x a ln x x a C x arccos Cx x2 a2 ax22 xa ax a22 2dx x2 a2)3ln x x2 a2 Cdxdx 1aClnxC2 x2 x 2 a dx (2x84a ln xx2 a2 C8xa arccosx含有 a2 x2 (a 0) 的積分dxa2 x2x arcsinadxxa2 a2 x22x 2 dx

7、22axdx(a2 x2)322 ax21a2 x22axarcsin C2aarcsin x C adxx a2 x2dxx2 a2 x21ln a a2 x2 a22ax2axa2x2dx(a2x2)3dxx a2x2dx x2 a2x2dxx8x(2x222x a2 x222 a32axarcsin2a 8x(5a2 2x2) a21x dx a2 x22 2 2 2 a x a x2 dx xx2)3 Cx2aln3a4 arcsin x C84axarcsin8aaa2 x2Cxarcsin Ca含有ax2 bx c (a 0) 的積分dx 1 ax2 bx c aln 2ax b

8、2 a ax2 bx c616263646566676869707172（九）7374ax2 bx2ax b 2cdx ax bx c4a4ac b28 a3ln 2ax b 2 a ax2 bx c C75xax2 bxdx 1 ax2 bx cab ln 2ax2 a3b 2 a ax2 bx c C76dx cbx2 ax77cbxax2dx78xdxcbx2 ax（十）含有xa或(x a)(b x) 的積分xb2ax b 2 c bx ax 4a12ax barcsin 2 C ab2 4acb2 4ac2ax barcsin C8 a3b2 4ac1 c bx ax2b 3 arcs

9、in 2a2x b Ca 2 a3b2 4ac82 (x798081b) xx ba (b a)ln(xab) bx ax (b a)arcsin bx xa C 2arcsin x a C (a b)22x a b (b a)2(x a)(b x)dx (x a)(b x) arcsin 4C(a b)十一）含有三角函數的積分83 sin xdx cosx Ccos xdx sinx Ccot xdx lnsinxCsec xdx lntan( x)C42csc xdx lntan xC ln2sec2 xdx tanxCcsc2 xdx cotxCCln cosxtan xdx ln se

10、cx tanxcscxsecxtanxdx secx Ccscxcotxdx cscx2 x 1sin xdx sin2x242 x 1 cos xdx sin2x24n 1 n 1sin xdx sin xcosxnn 1 n 1cos xdx cos xsinxn1 cosxn1n 1 sin x1 sin xdxnsin xdxn n 1 cos x n 1 cos xm n 1 cos xsin xdx mn1n n1nn2cotx Cn2sin xdxcosn 2 xdxdxn2sin xdxn2n 1 cos xn1n2m 1 n 1 m 1 cos xsin x n m n1 m

11、1 n 1 ncos xsin xm n m nm cos2nxsin xdxm n 2 cos xsin xdxsin ax cosbxdx cos(a b)x cos(a b)x C 2(a b) 2(a b)848586878889909192939495969798991001sin ax sin bxdx sin(a b)x2(a b)1cosaxcosbxdx sin(a b)x2(a b)1sin(a b)x C2(a b)1sin(a b)x2(a b)CCCab(a2arctan( tanx) ab aa2 cos2 x b2 sin2 xdxa2 cos2 x b2 sin

12、2 x1 ln2abxsin axdx 12 sinaxa22 1 2x sin axdx x cosax ax cosaxdx 12 cosaxa22 1 2x cosaxdx x sinax a 含有反三角函數的積分(其中btanx abtanx a1x cos ax C a22 x sin ax a21x sin axa22 x cos ax aa 0 )arcsinxdxxarcsin xa2 x2 Caa23 cosax a3b2)(a2(a2(ab2)b2)222 b2)23 sinax C a101102103104105106107108109110111112(十二)1132

13、ax)arcsin4ax22 axC4x2 arcsin xdx a3xx arcsin3a 19(x2a92a2) a2x2arccos xdx ax xarccosa2xx xarccos dx (a22ax)arccos4ax a2 x24x2 arccos xdx a3x arccos3a19(x2 2a2) ax2arctan xdx x arctan x aaln(a2a2x2) Cx arctan xdx 1(a2 a2x2 )arctan3x2 arctan dx arctana33a6 ln(ax2)含有指數函數的積分1axdx 1 ax Clnaax 1 axe dx e

14、Caax 1 xe dx 2 (ax a2n ax 1 n ax x e dx x e aax1)eax1 axe dxx x x xa dx a lna2a(ln a)2n x 1 n x a dx x ln aln an 1 xx a dxeax sin bxdx 2aeax(asinbx b cosbx )eax cosbxdx b22 1 2 eax (bsin bx a cosbx) ab114115116117118119120121(十三)1221231241251261271281292xx xarcsin dx (2（十五）含有雙曲函數的積分137shxdx chxC138c

15、hxdx shxC139thxdx lnchxC140sh2xdx x21sh2x C41412xch xdx 21sh2x C4（十六）定積分142cos nxdx sin nxdx 1430cosmxsinnxdx0130eax sin n bxdx1 ax n 1 2 2 2 e a b nsin bx(asinbxnbcosbx)131ax ne cos bxdx 2n(n 1)b2b2n2ax n 2 e sin bxdx1 a2axb2n2 encos1 bx(a cosbxnbsin bx)2n(n 1)b2bnaxe cosn 2bxdx132ln xdx xlnxxCdx133 lnln xCxln xn1n11134x ln xdx x1(lnxn1n135(ln x)n dxx(ln x)nn(ln x)136xm(ln x)ndx1m x1(ln x)n含有對數函