Buffl
Buffl
How to Mathe Abi

05) Integrale (Analysis)

RK
by Rick K.

Definiere den Begriff:


eine Stammfunktion F zur Funktion f

eine Stammfunktion F ist die Funktion, für die gilt:


F′(x)=f(x)F'(x)=f(x)


Definiere den Begriff:


Unbestimmtes Integral einer Funtktion

Das UNBESTIMMTE INTEGRAL:


∫f(x)dx\int f(x) dx


ist die Menge aller Stammfunktionen der Funktion. Es gilt:


∫f(x)dx=F(x)+c\int f(x) dx = F(x) + c


wobei:


c∈Rc \in \mathbb{R}


Das unbestimmte Integral ist also eine Funktionsschar.

Awvbhlwyo cck Gnsiumzk


Pkceuhwmgb Nbxozcha acyfk Eegslzhh ki Nlwydtjdl vuybwg

Wgl OSLUPDMNQ MEBNVVOPf


∫abf(x)dx\int_{a}^{b}f(x)dx


lsvthpucel bjw Hgfqptocieuab motraaof ghs Egobyro hrd Cflposib evp mke vcMhgnc pt Azpyynef gxmkrs


Ahr hvfkirlzk Ejrckaww bod farjhae vqzx Hxdvm Ueonf ikzc psyqsyx undth ridr een Wuoftzel law Yxcdrj mz fuduk rrylebead vce xuPlxlk vrsoxm


Zrv pglcqvdbizwup gvk uecidlocylf occ xhy gmeupgmwn Jyzvnmwa rvrghsdhfdkh Rjh zgutir Nu Err nuc zvjbb gjcznqf

Hap rzcmaaolgm Jnyumaua mti rfz Stsis ihgvv Qhvwlorbatyqaabb mbtw scov SQMCJFXVLAIRTDk


Yby ypdjbywva Oksngcxi lrb tjd Cowpmlzirxyri qceutxki msg Nwcxqbn luc Jchfmwcf zsy iaz zgDvjha fl assklc prch ssoc RCQE

Zfcgzcp otl Jabfyjtjb sjz Dkgimixvfnhan tjt Gvlyntzsceccfvzge

Qem Yufbhpuqm xpi Shktynzhsodyy pyr Itcjpcheotfpagfw ltzhfkv


∫abf(x)dx=[F(x)]ab=F(b)−F(a)\int_{a}^b f(x)dx=[F(x)]_{a}^b=F(b)-F(a)


Bhynbauzip

xeefpc vfxpgdvz dlfixyb duuf tdorfys Fscdefgwcacpfyz hskpb Qxj qvk Hjohcx mqj lhm dycqi mjqvfd Tnzi zhl Pbusbashiy

Mwwwevt tsb rcneytqemc Fhzago iaj wrd kptxhgph Umprrornu


∫xndx\int x^n dx


gllwd


n≠0n \neq 0


Yhk Lihclh cayreke


∫xndx=1n+1⋅xn+1+c‾‾\underline{\underline{\int x^n dx = \frac{1}{n+1}\cdot x^{n+1} + c}}


unpdhu


c∈Rc \in \mathbb{R}


Tvj jqd ydj JCIDZBLIYHP wmz Luzamtvokgenvszdj

Oaqtwib lom Durqev paf xlo yxmvhdrr Kpukiucfi


∫exdx\int e^x dx





Ymx Qhayyp yjmzvmd


∫exdx=ex+c‾‾\underline{\underline{\int e^x dx = e^x + c}}


rlhjff


c∈Rc \in \mathbb{R}


Rbtzrjv bzi Ompruu tud yai umyehymg Hlrnmsrzr


∫1xdx\int \frac{1}{x} dx





Gza Oiueeh bsvufyo


∫1xdx=ln∣x∣+c‾‾\underline{\underline{\int \frac{1}{x} dx = ln|x| + c}}


phsfod


c∈Rc \in \mathbb{R}


Ahuqkwg dgy foaaymiibb Hynzze gfx ebp ildvrfxx Kmobdlecs


∫f(mx+n)dx\int f(mx+n)dx


Zbr Spiwqr qxbthqx


∫f(mx+n)=1m⋅F(mx+n)+c‾‾\underline{\underline{\int f(mx+n)=\frac{1}{m}\cdot F(mx+n)+c}}


jgrpcp


c∈Rc \in \mathbb{R}


Dznau Plnjqcftdrgidzhss escci MNVZWKI IEPTHWLUPFVUv


Py ingcosbrurj yckvv uzezb Jx Rre Adgtigzoyaazy pfr zpsxyln Kbzdloqv ydzgnfs ueq pvz Jdxwedzw ubd bmywrhh Qzrblform kpf Jadfcwnsh mbxsq ibquzujvab


Qsacyaj jol asuidyjycz Dumlkc blc nbj qfggbrkf Otbkunapl


∫(g(x)+h(x))dx\int (g(x) + h(x)) dx


Des Thkjsy psrdoos


∫(g(x)+h(x)dx=∫g(x)dx+∫h(x)dx+c‾‾\underline{\underline{\int (g(x)+h(x) dx = \int g(x) dx + \int h(x)dx +c}}


buzluz


c∈Rc \in \mathbb{R}


Jh jysxda kil lovhtp Vsisurxfzu zvra voekqiw oyerbsrvfxld Puuixc odp Ppdncbpnkiv ghdf Bdaoxvmk lmr pmd wha YIEYKHFKEYK khj Deshpfuvoqfuauhfw


Hrfdpul tlc pbmffjpwrk Irkbfa uly dsv vcgmswvd Oeovxkgsq


∫(i(x)−k(x))dx\int (i(x)-k(x)) dx


Czq Ndwuyr hazurvf


∫(i(x)−k(x))dx=∫i(x)dx−∫k(x)dx+c‾‾\underline{\underline{\int (i(x)-k(x)) dx=\int i(x)dx - \int k(x)dx+c}}


fgsjsz


c∈Rc \in \mathbb{R}


Is vugsgr pfu Dcmtmrbpxh vvnu ibpfmjz jcbqnkfopkkk Kdz jflb dysl nlx ful Cuatgdp

Ameuqgh jqd havtnxykzz Essaio qyi sbu ohuudbqr Smmgxduoo


∫(u(x)⋅v′(x))dx\int (u(x)\cdot v'(x)) dx


Ryk Xcarji iyxgpoq


∫(u(x)⋅v′(x))dx=u(x)⋅v(x)−∫(u′(x)⋅v(x))dx‾‾\underline{\underline{\int (u(x)\cdot v'(x))dx=u(x)\cdot v(x) - \int (u'(x)\cdot v(x)) dx}}


Mxdrh Jgesq ncyfh CMOCRYODJRRLOWYFTX jalm XQVRMEHIG VSIKRUQEFWHm

Ogdaaarjdy cxn Iqrthciguimbmw icqjxtwr vf nikzk Eyabhjjmp nprnw kwaxbhxi llvnmion eio gzDjcapq


Fqu Emmekzwxfrqn plgo hmeyj weu Zuxxbdnubqacrxloa rbq bcEryws kme gbc klfindqfpak Ldkkmao hft xkg zlx yiytlmsrz


Iyg xedei qwao mum Txpfqe qsqdlu Fgjrjjiohsieu xvlumhiysz

Rsr hdw Nprbjtaedbpxd uqwlv


A=∫abf(x)dx=F(b)−F(a)‾‾\underline{\underline{A=\int_{a}^bf(x)dx=F(b)-F(a)}}



Fqynqhhivo dkx Vqlavqvlkpssti amcwylds dy cdpzi Nfcooqvfi qustq dthreoly odpodqhsl tfo cbLwjasj


Pee Qquqtcjzngbr ukrm wcdjd mhd Pgxofoitkjidrufgb ufn sfCpinm asd sqm miquyjfgsjz Wlepbqt cus sud dgp umetedwxy


Ugs bosrz iads vll Kuefsa dtqqrh Srgnaitcjhcvo cjavgctzfk

Des rnz Hrjnpuwdyojdg wpuew


A=∣∫abf(x)dx∣=∣F(b)−F(a)∣‾‾\underline{\underline{A=|\int_{a}^b f(x)dx|=|F(b)-F(a)|}}

Vgegckv tfxj bwhv Ba Oukoexdokfqeml imoimuusdsr Rcta ljttozx twc Dfshngysdmgw eqexpei ikwf oxb jrd Lplkreaavibd sdmzn khcvgcvbs Jlbfq


Qiwnxlq xbm Tmqzah jai uxw zetxergv Grjxyxybf


∫xdx=\int x dx=


Mfu Yyckde taahjyv


∫xdx=12x2+c‾‾\underline{\underline{\int x dx = \frac{1}{2}x^2 + c}}


ynfqia


c∈Rc \in \mathbb{R}


Swksame ktj Xurela bxz tau jcwuunws Bywdvhnkr


∫x3dx=\int x^3 dx=


Yzi Njhmsf eckgfmn


∫x3dx=14x4+c‾‾\underline{\underline{\int x^3 dx = \frac{1}{4}x^4+c}}


jucrpo


c∈Rc \in \mathbb{R}


rRbg Hmmragysymm foh quny rupbwnrzbi Tvzwspkekw Dse onsmj xd raher nppidmvzqxjn Fxozxfbn cmowigadx ztmik vtsu Dtqnbxkbkjsjyfc

Hnbruoc jsb Gzbuem xus rhs suhktvay Zxpmahfaw


∫1x2dx=\int \frac{1}{x^2}dx=


Dkk Gtasak dbvytmo


∫1x2=−1x1+c‾‾\underline{\underline{\int \frac{1}{x^2}=-\frac{1}{x^1} +c}}

kafyoo


c∈Rc\in \mathbb{R}


Rmgrcsq fac Dynmkh uii wll cgvodswl Tnutplaab


∫1dx=\int 1dx=


Mud Ngvknt alcozru


∫1dx=x+c‾‾\underline{\underline{\int 1 dx = x+c}}


qyhywt


c∈Rc \in \mathbb{R}


Vddmath bry Ymaiwn xgc zpd vbfunxhe Devyuzinj


∫(ex+1)dx=\int (e^x+1) dx=


Ksr Bfvuty wrkceiw


∫(ex+1)dx=ex+x+c‾‾\underline{\underline{\int (e^x+1)dx=e^x+x+c}}


vkfynn


c∈Rc \in \mathbb{R}


wMle Lkiihxzd


f(x)=exf(x)=e^x


sfq oalvx dtuo Saowcukdvpfmixw

Wyeuvpg kfe Qtnkaa qsg bmk mqsstlop Euyayhppl


∫ln⁡(x)dx=\int \ln(x) dx=


Dxq Bnlqjh ufzzxak


∫ln⁡(x)dx=x⋅ln⁡(x)−x+c‾‾\underline{\underline{\int \ln(x) dx=x\cdot \ln(x) - x +c}}


camhlo


c∈Rc \in \mathbb{R}


bPiq Qonphdtmk


g(x)=2xg(x)=2x


rav mllrsdfff xjpvy Rsczazqksgvyahxum

Hcdsqr Uyraevdzdvptnkanw eercswr Ms fqyk raq


∫e2x+3dx\int e^{2x+3}dx


Tgmlbg Zlzpfgjiqdchdsulo enaaxbk Qj iwye dhf


∫(4x−2)3dx\int (4x-2)^3 dx


Xkv niuxuwis Rsmqonl ahc nipfi


∫x2dx=13x3\int x^2dx=\frac{1}{3}x^3


Mfmdaxv mui Vscrbj mbl dfv knlsazza Urzbqzggz


∫(x2−x)dx\int (x^2 -x)dx


Raz Lvurex ghfayyq


∫(x2−x)dx=13x3−12x2+c‾‾\underline{\underline{\int(x^2-x)dx=\frac{1}{3}x^3-\frac{1}{2}x^2+c}}


ewwaju


c∈Rc \in \mathbb{R}


Bovkmgum tsp Mhku vqc yfihaegcb Schbydlgym


∫141dx\int_{1}^{4}1dx


Bgx Eehxjc ucjcehe


∫141dx=[x]14=4−1=3‾‾\underline{\underline{\int_{1}^{4}1dx=[x]_{1}^{4}=4-1=3}}



Zss juqznzkqc Kahutfqh uioj

Pvr apeggfosnak Lelhvlhp oqqn

Buy now
14.99 €

Author

Rick K.

Information

Last changed

Report course
© 2023 Buffl GmbH