常用积分表

常 用 积 分 公 式（一）含有的积分() ax b +0a ≠1．d x ax b +∫＝1ln ax b C a ++2．＝()ax b x μ+∫d 11()(1)ax b C a μμ++++（1μ≠−）3．d x x ax b +∫＝21(ln )ax b b ax b C a +−++4．2d x x ax b +∫＝22311()2()ln 2ax b b ax b b ax b C a ⎡⎤+−++++⎢⎥⎣⎦5．d ()x x ax b +∫＝1ln ax bC b x +−+6．2d ()x x ax b +∫＝21ln a ax bC bx b x+−++ 7．2d ()x x ax b +∫＝21(ln )bax b C a ax ++++b8．22d ()x x ax b +∫＝231(2ln b ax b b ax b C a ax b +−+−++9．2d ()x x ax b +∫＝211ln ()ax b C b ax b b x+−++的积分10．x ∫C +11．x ∫＝22(3215ax b C a −+12．x x ∫＝22232(15128105a x abx b C a−++13．x∫＝22(23ax b C a −+14．2x ∫＝22232(34815a x abx b C a −++15．∫＝(0)(0)C b C b ⎧+>+<16．∫2a bx b −−∫17．d x x ∫＝b + 18．2d xx ∫＝2a x −+ （三）含有22x a ±的积分 19．22d x x a +∫=1arctan xC aa +20．22d ()n x x a +∫=2221222123d 2(1)()2(1)()n n x n n a x a n a x a −−x−+−+−+∫21．22d x x a −∫=1ln 2x a C a x a−++（四）含有的积分2(0ax b a +>)22．2d x ax b +∫＝(0)(0)x C b Cb ⎧+>⎪⎪⎨+<23．2d x x ax b +∫＝21ln 2ax b C a ++24．22d x x ax b +∫＝2d x b xa a axb −+∫25．2d ()x x ax b +∫＝221ln 2x C b ax b++26．22d ()x x ax b +∫＝21d a xbx b ax b −−+∫ 27．32d ()x x ax b +∫＝22221ln 22ax b a C b x bx+−+ 28．22d ()x ax b +∫＝221d 2()2x xb ax b b ax b +++∫（五）含有的积分2ax bx c ++(0a >)29．2d x ax bx c ++∫＝22(4)(4)C b C b ac +<+>ac 30．2d x x ax bx c ++∫＝221d ln 22b x ax bx c a a ax bx c++−++∫(0a >)的积分31．∫＝1arshxC a +＝ln(x C ++ 32．∫C +33．x ∫C34．x ∫＝C +35．2x ∫2ln(2a x −++C36．2x ∫＝ln(x C +++37．∫1ln aC a x −+38．∫2C a x −+40．x ∫＝2243(25ln(88x x a a x C ++++43．d x x ∫ln a a C x −++44．2d x x∫＝ln(x C x −+++(0a >)的积分45．＝1arch x xC x a+=C + 46．∫C +47．x ∫C +48．x ∫＝C +49．2x ∫22a ++C50．2x ∫＝ln C ++51．∫1arccos aC a x+52．∫2C a x +53．x ∫2ln 2a −+C54．x ∫＝2243(25ln 88x x a a C −++55．x ∫C +56．xx ∫＝422(288x a x a C −−+57．d x x∫arccos aa C x −+58．2d x x ∫＝ln C x −++(0a >)的积分 59．∫＝arcsinxC a + 60．∫C +61．x ∫＝C +62．x ∫C +63．2x ∫＝2arcsin 2a x C a ++64．2x ∫C +65．∫1ln a C a x −+66．∫2C a x −+67．∫x 2arcsin 2a C a++x68．∫x ＝2243(52arcsin 88x x a x a a C −++69．∫x ＝C +70．x∫x ＝422(2arcsin 88x a x x a C a−++71．d x x ∫ln a a C x −++72．2d x x∫＝arcsin xC x a −−+(0a >)的积分73．∫C +74．x ∫2C ++75．x ∫C −+76．∫＝C +77．x ∫2C ++78．x ∫＝C ++79．x ∫＝((x b b a C −−++80．x ∫＝((x b b a C −−+81．∫C+()a b <82．x ∫C ++()a b <（十一）含有三角函数的积分 83．sin d x x ∫＝cos x C −+84．cos d x x ∫＝sin x C + 85．tan d x x ∫＝ln cos x C −+ 86．cot d x x ∫＝ln sin x C + 87．sec d x x ∫＝ln tan()42xC π++＝ln sec tan x x C ++ 88．csc d x x ∫＝ln tan2xC +＝ln csc cot x x C −+ 89．2sec d x x ∫＝tan x C + 90．2csc d x x ∫＝cot x C −+ 91．sec tan d x x x ∫＝sec x C + 92．csc cot d x x x ∫＝csc x C −+93．2sin d x x ∫＝1sin 224x x C −+ 94．2cos d x x ∫＝1sin 224x x C ++95．sin d n x x ∫＝1211sin cos sin d n n n x x x n n−−−−+∫x 96．cos d n x x ∫＝1211cos sin cos d n n n x x x n n−−−+∫x 97．d sin n x x ∫＝121cos 2d 1sin 1sin n n x n xn x n −−−−⋅+−−∫x 98．d cos n x x ∫＝121sin 2d 1cos 1cos n n x n xn x n −−−⋅+−−∫x99．cos sin d m n x x x ∫＝11211cos sin cos sin d m n m nm x x x m n m n−+−x x −+++∫ ＝11211cos sin cos sin d m n m n n x x x m n m n +−−x x −−+++∫100．＝sin cos d ax bx x ∫11cos()cos()2()2()a b x a b x C a b a b −+−−++−101．＝sin sin d ax bx x ∫11sin()sin()2()2()a b x a b x C a b a b −++−++−102．＝cos cos d ax bx x ∫11sin()sin()2()2()a b x a b x C a b a b ++−++−103．d sin xa b x +∫tanxa b C ++22()a b >104．d sin x a b x +∫C+22()a b <105．d cos xa b x +∫)2x C +22()a b >106．d cos x a b x +∫C +22()a b <107．2222d cos sin x a x b x +∫＝1arctan(tan )bx C ab a + 108．2222d cos sin x a x b x 1tan ln 2tan b x a C ab b x a +−∫＝+−109．sin d x ax x ∫＝211sin cos ax x ax C a a −+ 110．2sin d x ax x ∫＝223122cos sin cos x ax x ax ax C a a a −+++111．cos d x ax x ∫＝211cos sin ax x ax C a a ++112．2cos d x ax x ∫＝223122sin cos sin x ax x ax ax C a a a+−+（十二）含有反三角函数的积分（其中）0a >113．arcsin d x x a ∫＝arcsin xx C a+114．arcsin d xx x a ∫＝C +115．2arcsin d xx x a∫＝3221arcsin (239x x x a C a ++116．arccos d xx a ∫＝arccosxx C a−+117．arccos d xx x a ∫＝C +118．2arccos d xx x a∫＝3221arccos (239x x x a C a −++ 119．arctand x x a ∫＝22arctan ln()2x a x a x C a −++ 120．arctan d x x x a∫＝221()arctan 22x aa x x C a +−+121．2arctan d xx x a∫＝33222arctan ln()366x x a a x a x C a −+++ （十三）含有指数函数的积分122．＝d xa x ∫1ln xa C a + 123．e d axx ∫＝1e ax C a +124．e d axx x ∫＝21(1)e ax ax C a −+125．e d n axx x ∫＝11e e n ax n ax n d x x x a a−−∫126．d xxa x ∫＝21ln (ln )x xx a a a a C −+ 127．d nxx a x ∫＝11d ln ln n x n xn x a x a a a −−∫x 128．＝e sin d axbx x ∫221e (sin cos )ax a bx b bx C a b −++ 129．＝e cos d ax bx x ∫221e (sin cos )axb bx a bx C a b+++130．＝e sin d ax n bx x ∫12221e sin (sin cos )ax n bx a bx nb bx a b n−−+ 22222(1)e sin d ax n n n b bx x a b n−−++∫ 131．＝e cos d ax n bx x ∫12221e cos (cos sin )ax n bx a bx nb bx a b n−++ 22222(1)e cos d ax n n n b bx x a b n−−++∫ （十四）含有对数函数的积分132．ln d x x ∫＝ln x x x C −+ 133．d ln x x x ∫＝ln ln x C +134．ln d n x x x ∫＝111(ln )11n x x C n n +−+++ 135．(ln )d n x x ∫＝1(ln )(ln )d n n x x n x −−∫x 136．(ln )d m n x x x ∫＝111(ln )(ln )d 11m n m n n x x x x m m +−−++∫x （十五）含有双曲函数的积分137．sh d x x ∫＝ch x C +138．ch d x x ∫＝sh x C +139．th d x x ∫＝ln ch x C + 140．2sh d x x ∫＝1sh224x x C −++ 141．2ch d x x ∫＝1sh224x x C ++ （十六）定积分142．＝＝0 cos d nx x π−π∫sin d nx x π−π∫143．＝0 cos sin d mx nx x π−π∫144．＝ cos cos d mx nx x π−π∫0,,m n m n ≠⎧⎨π=⎩145．＝ sin sin d mx nx x π−π∫0,,m n m n ≠⎧⎨π=⎩146．＝＝0sin sin d mx nx x π∫0cos cos d mx nx x π∫0,,2m n m n ≠⎧⎪⎨π=⎪⎩ 147． n I ＝20sin d n x x π∫＝20cos d n x x π∫ n I ＝21n n I n−− 134225n n n I n n −−=⋅⋅⋅⋅−"3（为大于1的正奇数），n 1I ＝1 13312422n n n I n n −−π=⋅⋅⋅⋅⋅"n （为正偶数），0I ＝2π −。

1 2
u 2x
2x e d (2 x )


1 u e du 2 1 u e C 2 1 2x e C 2
一般情况下： 设 f (u) 有原函数 F (u) , 即 F ' (u) f (u) f (u)du F ( u) C
F ( u ) F [ ( x )] 若u ( x )可导 d F [ ( x )] F ' ( u) ' ( x ) dx f ( u) ' ( x ) f [ ( x )] ' ( x ) F[ ( x )]是 f [ ( x )] ' ( x ) 的原函数 f [ ( x )] ' ( x )dx F [ ( x )] C F ( u) C f ( u)du 这样, 我们就得到下面的定理 :
例20

x (1 x )3 dx . x 11 (1 x )3 dx
例3
sec
2
2
( 3 x 4)dx


1 3
1 sec (3 x 4) d (3 x 4) 3

sec2 (3 x 4) d (3 x 4)
1 3
令u 3 x 4

se c2 u du

1 tan u C 3 1 tan (3 x 4) C 3
例4
x
1 x dx
2
2

1 x
1 2 1 x 2 d (1 x 2 ) 1 令u 1 x 2 u du 2 3 1 2 2 2 3u C 3 3 1 2 1 2 2 u C (1 x ) C 3 3

常用积分表（Common integral table）Common integral formula(a) containing the integral (axb+0a =) Formula 1.dxaxb+ = 1lnaxbCa++2. = (axbx) + 11 (formula)(1)AxbCa + + + Mu+(U = 1.)3.dxxaxb+ = formula21 (LN) axbbaxbCa+.++The 4.2dxxaxb+ formula (2) = 22311(ln2axbbaxbbaxbCa..+.++++.)...Five(d)Formula xxaxb+ = 1lnaxbCbx+16.2d ()Xxaxb+ = formula21lnaaxbCbxbx+Type7.2d ()Xxaxb+ = formula21 (LN) baxbCaax++++8.22d ()Xxaxb+ = formula231 (2ln) baxbbaxbCaa+.+.+ 9.2d ()Xxaxb+ = formula211ln ()AxbCbaxbbx++(two) an integral containing axb+10.daxbx+ formula (= 32)3axbCa++Formula 11.dxaxbx+ = 322 (32) ()15axbaxbCa.++Formula 12.2dxaxbx+ = 222332 (15128) () 105axabxbaxbCa.+++13.dxxaxb+ = formula22 (2)3axbaxbCa.+Formula 14.2dxxaxb+ = 22232 (348)15axabxbaxbCa.++FifteenDxxaxb+ = formula1ln (0)2arctan (0) AxbbCbbaxbbaxbCbbb.+. +>.++..1+<....16.2dxxaxb+ = formula D2axbaxbxbxaxb+.+ formula17.daxbxx+ = formula D2xaxbbxaxb+++ formula18.2daxbxx+ = formulaD2axbaxxxaxb+1+ formula(three) an integral containing 2xa +19.22dxxa+ formula =1arctanxCaa+20.22d (nxxa+) formula =2221222123d2 (1) (2) (1) (nnxnnaxanaxa..).+.+.+ formula21.22dxxa. formula =1ln2xaCaxa.++(four) contains integral 2 (0axba+>22.2dxaxb+ = formula1arctan (0)1ln (0)2axCbbabaxbCbabaxb.+>...+<..+.....Formula 23.2dxxaxb+ = 21ln2axbCa++ 24.22dxxaxb+ = formula2dxbxaaaxb.+ formula25.2d ()Xxaxb+ = formula221ln2xCbaxb++26.22d ()Xxaxb+ = formula21daxbxbaxb..+ formula27.32d ()Xxaxb+ = formula22221ln22axbaCbxbx+128.22d ()Xaxb+ = 221d2 (2xxbaxbbaxb+) formula+ + formula(five) contains integral 2axbxc++ (0a>)29.2dxaxbxc++ = formula222222222arctan (4)44124ln (4)424axbCbacbacbaxbbacCbacbacaxbbac+.+<...+> +.....++..Formula 30.2dxxaxbxc++ = 221dln22bxaxbxcaaaxbxc++.+ + formula(six) contains an integral of 22xa+ (0a>) Formula 31.22dxxa+ = 1arshxCa+ = 22ln (xxaC+++) 32.223d ()Xxa+ = formula222xCaxa++Formula 33.22dxxxa+ = 22xaC++34.223d ()Xxxa+ = formula221Cxa.++35.222dxxxa+ = formula22222ln ()22xaxaxxa+.+++36.2223d ()Xxxa+ = 2222ln (xxxaCxa.++++) formula+37.22dxxxa+ = formula221lnxaaCax+.+38.222dxxxa+ = formula222xaCax+139.22dxax+ = formula22222ln ()22xaxaxxa+++++(40.223) xax+ (25) ln formula = 22224223 () 88xxaxaaxxaC++++++41.22dxxa+ formula (= 2231)3xaC++42.222dxxa+ = formula4222222 (2) ln ()88xaxaxaxxaC++.+++43.22dxaxx+ = formula2222lnxaaxaaCx+.+ + +44.222dxaxx+ = formula2222ln () xaxxaCx+.++++(seven) contains an integral of 22xa. (0a>) Formula 45.22dxxa. = 1archxxCxa+=22lnxxaC+.+ 46.22d ()Xxa. = formula222xCaxa.+.Formula 47.22dxxxa. = 22xaC.+48.223d ()xxxa.∫＝221cxa. +.49.222dxxxa.∫＝ln22xaxaxxa 22222. + + +50.2223d ()xxxa.∫＝2222lnxxxacxa. + +. + .51.22dxxxa.∫＝1arccosacax +52.222dxxxa.∫＝222xacax.+53.22dxax.∫＝ln22xaxaxxa 22222 +... +.54.223 () xax.∫＝22224223 (25) ln88xxaxaaxxac. + + +55.22dxxax.∫＝2231 ()3xac. +56.222dxxax.∫＝4222222 (2) ln88xaxaxaxxac +... +.57.22dxaxx.∫＝22arccosaxaacx. +58.222dxaxx.∫＝2222lnxaxxacx.. + + +(八) 含有22ax. (0a > 的积分59.22dxax.∫＝arcsinxca +60.22d ()xax.∫＝222xcaax +.61.22dxxax.∫＝22axc. +62.223d ()xxax.∫＝221cax +.63.222dxxax.∫＝222arcsin22xaxaxca. + + 64.2223d ()xxax.∫＝22arcsinxxcaax. +.65.22dxxax.∫＝221lnaaxcax...+66.222dxxax.∫＝222axcax.. +67.22dax.∫＝222arcsin22xaaxca. + +68.223 () ax.∫＝222243 (52) arcsin88xxaxaxaa. + +69.22dxax.∫＝2231 ()3axc. +70.222dxax.∫＝4-2222. no, 2222 (2) arcsin88xaxaaxca. + +71.22daxxx.∫＝2222lnaaxaxacx.... + +72.222daxxx.∫＝22arcsinaxxcxa.. +(九) 含有2axbxc± + + (0a > 的积分73.2dxaxbxc ∫＝21ln22axbaaxbxcca + + + + + + +74.2daxbxcx + + ∫＝224axbaxbxca ++ + + + + +2234ln228acbaxbaaxbxcca.+ + + + + +75.2dxxaxbxc + + + + ∫＝21axbxca23ln222baxbaaxbxcca. + + + + +76.2dxcbxax +.∫＝212arcsin4axbcabac.. ++77.2dcbxaxx +.∫＝223224arcsin484axbbacaxbcbxaxcaabac. ++ + +.+78.2dxxcbxax +.∫＝23212arcsin24baxbcbxaxcaabac. . + + +.+(十) 含有xaxb.+.或 () (xabx..的积分79.dxaxxb..∫＝ () () ln () xaxbbaxaxbxb.. +... +. +.80.dxaxbx..∫＝ () () arcsinxaxaxbbabxbx.... +. +...81.d () (xxabx..∫＝2arcsinxacbx.+.() ab <82. () () dxabxx..∫＝22 () () () arcsin44xabbaxaxabxcbx.... . + +.() ab <(十一) 含有三角函数的积分83.sindxx∫＝cosxc. +84.cosdxx∫＝sinxc +85.tandxx∫＝lncosxc. +86.cotdxx∫＝lnsinxc +87.secdxx∫＝lntan ()42xcπ + + + + ＝lnsectanxxc88.cscdxx∫＝lntan2xc + ＝lncsccotxxc. +89.2secdxx∫＝tanxc +90.2cscdxx∫＝cotxc. +91.sectandxxx∫＝secxc +92.csccotdxxx∫＝cscxc. +93.2sindxx∫＝1sin224xxc. +94.2cosdxx∫＝1sin224xxc + +95.sindnxx∫＝1211sincossindnnnxxxnn... . + ∫96.cosdnxx∫＝1211cossincosdnnnxxxnn... + ∫97.dsinnxx∫＝121cos2d1sin1sinnnxnxnxn..... +..∫98.dcosnxx∫＝121sin2d1cos1cosnnxnxnxn..... +..∫99.cossindmnxxx∫＝11211cossincossindmnmnmxxxmnmn. +. ++ + ∫＝11211cossincossindmnmnnxxxmnmn +.".+ + ∫100.＝sincosdaxbxx∫11cos cos () ()(2) (2)abxabxcabab. + ++.the 101.＝sinsindaxbxx∫11sin () () (2) (2)abxabxcabab. + + +.+.the 102.＝coscosdaxbxx∫11sin () () (2) (2)abxabxcabab + + +.+.103.dsinxabx + ∫＝2222tan22arctanxabcabab ++...(ab)104.dsinxabx + ∫＝222222 tan12lntan2xabbacxbaabba +. ++ +.("),105.dcosxabx + ∫＝2arctan (tan)2ababxcababab +.++ +.(ab)106.dcosxabx + ∫＝tan12lntan2xababbacabbaxabba ++ + +.+ +..("),107.2222dcossinxaxbx＝108.2222dcossinxaxbx＝109.sindxaxx∫＝211sincosaxxaxcaa.110.2sindxaxx∫＝223122cossincosxaxxaxaxcaaa. + + 111.cosdxaxx∫＝211cossinaxxaxcaa + +112.2cosdxaxx∫＝223122sincossinxaxxaxaxcaaa + +. (十二) 含有反三角函数的积分 (其中) 0a.113.arcsindxxa∫＝22arcsinxxaxca + +.114.arcsindxxxa∫＝() other arcsin244xaxxaxca. +.115.2arcsindxxxa∫＝322221arcsin (2)39xxxaaxca + + +.116.arccosdxxa∫＝22arccosxxaxca.. +117.arccosdxxxa∫＝() other arccos244xaxxaxca... "118.2arccosd xxxa∫＝322221arccos (2)39xxxaaxca. + +.119.arctandxxa∫＝22arctanln ()2xaxaxca. + +120.arctandxxxa∫＝221 () arctan22xaaxxca +. 121.2arctandxxxa∫＝33222arctanln ()366xxaaxaxca. + + +含有指数函数的积分 (十三)122.＝dxax∫1lnxaca +123.edaxx∫＝1eaxca +124.edaxxx∫＝21 (1) eaxaxca.125.ednaxxx∫＝11eenaxnaxnxxxaa..∫126.dxxax∫＝21ln (ln)xxxaaaa.127.dnxxax∫＝11dlnlnnxnxnxaxaaa..∫128.＝esindaxbxx∫221e (sincos) axabxbbxcab.+129.＝ecosdaxbxx∫221e (sincos) axbbxabxcab + ++130.＝esindaxnbxx∫12221esin (sincos) axnbxabxnbbxabn. +(1) esindaxnnnbbxxabn 22222.+"∫131.＝ecosdaxnbxx∫12221ecos (cossin) axnbxabxnbbxabn. +(1) ecosdaxnnnbbxxabn 22222.+"∫含有对数函数的积分 (十四)132.lndxx∫＝lnxxxc. +133.dlnxxx∫＝lnlnxc +134.lndnxxx∫＝111 (ln)11nxxcnn + +.+ + + + +(nl) dnxx∫＝ 135.1 (in) (the) dnnxxnx..∫136. ('") (in) (the) dmnxxx∫＝111 d11mnmnnxxxxmm +. + + ∫含有双曲函数的积分 (十五)137.shdxx∫＝chxc +138.chdxx∫＝shxc +139.thdxx∫＝lnchxc +140.2shdxx∫＝1sh224xxc. + +141.2chdxx∫＝1sh224xxc + +定积分 (十六)142.＝＝0 cosdnxxπ.π∫sindnxxπ.π∫143.＝0 cossindmxnxxπ.π∫144.＝ coscosdmxnxxπ.π∫0..mnmn≠.d =.145.＝ sinsindmxnxxπ.π∫0..mnmn≠.d =.146.＝＝0sinsindmxnxxπ∫0coscosdmxnxxπ∫0,, 2mnmn≠...d =.ni＝20sindnxxπ∫＝20cosdnxxπ∫ 147. ni＝21nnin..134225nnninn..=...... (为大于1的正奇数), n1i＝113312422 (为正偶数), 0i＝2π。

8
8
a
∫ 71． a2 − x2 dx ＝ a2 − x2 + a ln a − a2 − x2 + C
x
x
∫ 72．
a2 − x2
x2
dx
＝
−
a2 − x2 − arcsin x + C
x
a
（九）含有 ±ax2 + bx + c (a > 0) 的积分
∫ 73．
dx
＝ 1 ln 2ax + b + 2 a ax2 + bx + c + C
常用积分公式
（一）含有 ax + b 的积分( a ≠ 0 )
1． ∫
dx ax +
b
＝
1 a
ln
ax
+
b
+
C
∫ 2． (ax + b)μdx ＝ 1 (ax + b)μ+1 + C （ μ ≠ −1 ）
a(μ + 1)
∫ 3． x dx ＝ 1 (ax + b − b ln ax + b ) + C ax + b a2
x2 ax2 + b
+C
∫ ∫ 26．
dx ＝ − 1 − a x2 (ax2 + b) bx b
dx ax2 + b
∫ 27．
dx x3(ax2 + b)
＝
a 2b2
ln
ax2 + b x2
−
1 2bx2
+C
∫ ∫ 28．
dx ＝
x