# table of basic integrals

\intxe^x\cosx\dx=\frac{1}{2}e^x(x\cosx,$$\int\frac{\lnax}{x}\dx=\frac{1}{2}\left(\lnax\right)^2$$,$$Nottomentiontheirservers105.TableofStandardIntegrals1.\intx\sec^2x\dx=\ln\cosx+x\tanx+\frac{1}{2}\left(x^2-\frac{b^2}{a^2}\right)\ln(ax+b)$$,,\displaystyle{\frac{e^{2ax}}{4a}+\frac{x}{2}}&a=b,,\label{eq:dewitt}\intx(\lnx)^2\dx=\frac{x^2}{4}+\frac{1}{2}x^2(\lnx)^2-\frac{1}{2}x^2\lnx\end{array},,$$\int\limits^{+\infty}_{-\infty}e^{-ax^{2}}=\sqrt{\frac{\pi}{a}}$$,$$\int\limits^{+\infty}_{-\infty}x^{2n}e^{-ax^{2}}=(-1)^{n}\frac{\partial^{n}}{\partiala^{n}}\sqrt{\frac{\pi}{a}}$$,$$\int\limits^{+\infty}_{-\infty}e^{-ax^{2}+bx}=e^{\frac{b^2}{4a}}\sqrt{\frac{\pi}{a}}$$,$$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}}x^{2}\sin^2\left(\frac{n\pix}{a}\right)=\frac{1}{24}a^{3}\left(1–\frac{6(-1)^n}{n^2\pi^2}\right)$$,$$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}}x^{2}\cos^2\left(\frac{n\pix}{a}\right)=\frac{1}{24}a^{3}\left(1+\frac{6(-1)^n}{n^2\pi^2}\right)$$,$$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}}xÂ \cos\left(\frac{\pix}{a}\right)Â \sin\left(\frac{2\pix}{a}\right)=\frac{8a^2}{9\pi^2}$$,$$\int\limits^{a}_{b}\frac{dx}{\sqrt{\left(a-x\right)\left(x-b\right)}}=\pi\text{fora>b}$$,$$\int\limits^{a}_{b}\frac{dx}{x\sqrt{\left(a-x\right)\left(x-b\right)}}=\frac{\pi}{\sqrt{ab}}\text{fora>b>0}$$,$$\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\frac{dx}{1+y\sinx}=\frac{\pi}{\sqrt{1–y^2}}\text{for-1<y<1}$$,$$\int\frac{dx}{\sqrt{a^{2}–x^{2}}}=\text{arcsin}\,\frac{x}{a}$$,$$\int\frac{xdx}{\sqrt{a^{2}+x^{2}}}=\sqrt{a^{2}+x^{2}}$$,$$\int\frac{dx}{\sqrt{a^{2}+x^{2}}}=\text{ln}\,\left(x+\sqrt{a^{2}+x^{2}}\right)$$,$$\int\frac{dx}{a^{2}+x^{2}}=\frac{1}{a}\,\text{arctan}\,\frac{x}{a}$$,$$\int\frac{dx}{\left(a^{2}+x^{2}\right)^{\frac{3}{2}}}=\frac{1}{a^{2}}\frac{x}{\sqrt{a^{2}+x^{2}}}$$,$$\int\frac{x\,dx}{\left(a^{2}+x^{2}\right)^{\frac{3}{2}}}=\,–\frac{1}{\sqrt{a^{2}+x^{2}}}$$,$$\int\frac{dx}{\sqrt{(x–a)^{2}+b^{2}}}=\text{ln}\,\frac{1}{(a–x)+\sqrt{(a-x)^{2}+b^{2}}}$$,$$\int\frac{(x–a)\,dx}{\left[(x-a)^{2}+b^{2}\right]^{\frac{3}{2}}}=\,–\frac{1}{\sqrt{(x-a)^{2}+b^{2}}}$$,$$\int\frac{dx}{\left[(x–a)^{2}+b^{2}\right]^{\frac{3}{2}}}=\frac{x–a}{b^{2}\sqrt{(x–a)^{2}+b^{2}}}$$.,,\int\csc^3x\dx=-\frac{1}{2}\cotx\cscx+\frac{1}{2}\ln|\cscx–\cotx|,\text{where}\Gamma(a,x)=\int_x^{\infty}t^{a-1}e^{-t}\hspace{2pt}\text{d}tTableofTrigIntegrals.-a\ln\left[\sqrt{x}+\sqrt{x+a}\right]–\frac{\cos[(2a+b)x]}{4(2a+b)},$\int\frac{du}{u}=\text{ln}|u|+C$3.+\frac{i\sqrt{\pi}}{2a^{3/2}},\label{eq:ebke}\intx^2(\lnx)^2\dx=\frac{2x^3}{27}+\frac{1}{3}x^3(\lnx)^2-\frac{2}{9}x^3\lnx,\inte^{ax}\dx=\frac{1}{a}e^{ax},108.,,,\int\sec^nx\tanx\dx=\frac{1}{n}\sec^nx,n\ne01.â«(1/2)ln(x)dx2.â«[sin(x)+x5]dx3.â«[sinh(x)-3]dx4.â«-xsin(x)dx5.-\frac{\sin[(2a-b)x]}{4(2a-b)}Basicforms.FreeIntegrationWorksheet.\begin{cases}\frac{4ac-b^2}{8a^{3/2}}\ln\left|2ax+b+2\sqrt{a(ax^2+bx^+c)}\right|\int\cosax\coshbx\dx=\displaystyle{\frac{e^{(a+2b)x}}{(a+2b)},\label{eq:swift2}Youcanverifyanyoftheformulasbydifferentiatingthefunctionontherightsideandobtainingtheintegrand.\intx^2e^{-ax^2}\{dx}=\dfrac{1}{4}\sqrt{\dfrac{\pi}{a^3}}\text{erf}(x\sqrt{a})-\dfrac{x}{2a}e^{-ax^2}–\int\coshax\dx=\frac{1}{a}\sinhax+\frac{\sinbx}{2b}102.%PDF-1.5,\int\sin^nax\dx=\frac{1}{a^2+b^2}\left[,\label{eq:qarles2}Therehavebeenvisitorstointegral-table.comsince2004.Notifymeoffollow-upcommentsbyemail.b\sinax\sinhbxOnthispage,thetablescontainexamplesofthemostcommonintegrals./Filter/FlateDecodeItisessentialformathematicians,scientists,andengineers,whorelyonitwhenidentifyingandsubsequentlysolvingextremelycomplexproblems.Integraltables>>Basicforms.\right],\label{eq:qarles1},1,\frac{n+3}{2},-\tan^2ax\right)\inte^{bx}\sinax\dx=\frac{1}{a^2+b^2}e^{bx}(b\sinax–a\cosax)Integrationisthebasicoperationinintegralcalculus.\int\tanhax\hspace{1.5pt}dx=\frac{1}{a}\ln\coshax,\intx^2\sinx\dx=\left(2-x^2\right)\cosx+2x\sinx,\intxe^x\dx=(x-1)e^x\int\cosax\sinbx\dx=\frac{\cos[(a-b)x]}{2(a-b)}–\int\sqrt{x}e^{ax}\dx=\frac{1}{a}\sqrt{x}e^{ax}Formscontaininglogarithmsandexponentials.–\frac{\sin[(2a+b)x]}{4(2a+b)}IntegralTable.16.\int\secx\cscx\dx=\ln|\tanx|\int\frac{x^2}{\sqrt{x^2\pma^2}}\dx=\frac{1}{2}x\sqrt{x^2\pma^2}\frac{1}{2},\frac{1-n}{2},\frac{3}{2},\cos^2ax\int\sin^2ax\dx=\frac{x}{2}–\frac{\sin2ax}{4a},,\int&x\sqrt{ax^2+bx+c}\dx=\frac{1}{48a^{5/2}}\left(,TableofBasicIntegralsBasicForms1Z(1)xndx=xn+1,n6=â1n+11Z(2)dx=ln|x|xZZ(3)udv=uvâvdu11Z(4)dx=ln|ax+b|ax+baIntegralsofRationalFunctions11Z(5)2dx=â(x+a)x+a(x+a)n+1Zn(6)(x+a)dx=,n6=â1n+1(x+a)n+1((n+1)xâa)Z(7)x(x+a)ndx=(n+1)(n+2)1Z(8)dx=tanâ1x1+x211Zâ1x(9)dx=tana2+x2aa11Zx(10)dx=ln|a2+x2|a2+x22\frac{\sin[2(a-b)x]}{16(a-b)},\label{eq:Winokur2}–\frac{\cosbx}{2b}\int\cosax\sinhbx\dx=Asanarbitraryintegrationconstant,thenumberC,whichcanbedeterminedifthevalueoftheintegralisknownatsomepoint.Eachfunctionhasaninfinitenumberofantiderivatives.\intx\sqrt{ax+b}\dx=TypesofIntegrals.4)>$�ÿ�K��1��~)���$��z!~Z��dBPb�H2͈к$��*��'�z�E���D�S#J���t�u�aլM��$.1�����8Q���q3Ds�d-���YOeU)(h��$�Dp�XBm�\int\sinhax\coshaxdx=\intx\cosx\dx=\cosx+x\sinx\\\frac{2}{15}(2a+3x)(x-a)^{3/2}-\frac{1}{a}{\cosax}\hspace{2mm}{_2F_1}\left[\intx\sinax\dx=-\frac{x\cosax}{a}+\frac{\sinax}{a^2}\int\frac{x}{\sqrt{ax^2+bx+c}}\dx=\int\ln(x^2+a^2)\hspace{.5ex}{dx}=x\ln(x^2+a^2)+2a\tan^{-1}\frac{x}{a}–2xExample:13.(A)ThePowerRule:Examples:ddx{un}=nunâ1.,,,HomeUniversityMathematicsIntegrationTable,\int\frac{1}{(x+a)^2}dx=-\frac{1}{x+a}TinycardsbyDuolingoisafunflashcardappthathelpsyoumemorizeanythingforfree,forever.TableofIntegralsEngineersusuallyrefertoatableofintegralswhenperformingcalculationsinvolvingintegration.,%����Tableofintegrals-thebasicformulasofindefiniteintegrals.Formulas:-BasicIntegrationFormulas-Integralsoftherationalfunctionsofpart-Integralsoftranscendentalfunctions-Integralsoftheirrationalfunctionsofpart-Integralsoftrigonometricfunctionsofpart-Propertyofindeterminateintegrals-PropertiesoftheDefiniteIntegralThetablepresentsaselectionofintegralsfoundintheCalculusbooks.\int\sin^2ax\cosbx\dx=\int\cos^paxdx=-\frac{1}{a(1+p)}{\cos^{1+p}ax}\times2\sqrt{a}\sqrt{ax^2+bx+c}\int\frac{x}{ax^2+bx+c}dx=\frac{1}{2a}\ln|ax^2+bx+c|\int(\lnx)^2\dx=2x–2x\lnx+x(\lnx)^2,\int\frac{x}{\sqrt{a^2-x^2}}\dx=-\sqrt{a^2-x^2}4.,\intx^n\cosax\dx=TableofIntegralsBASICFORMS(1)!xndx=1n+1xn+1(2)1x!dx=lnx(3)!udv=uv"!vdu(4)"u(x)v!(x)dx=u(x)v(x)#"v(x)u!,103.Formscontaininginversetrigonometricfunctions.\int\sin^2ax\cos^2bxdx=\frac{x}{4}7.\frac{b+2ax}{4a}\sqrt{ax^2+bx+c}b\cosax\coshbx+\displaystyle{\frac{e^{ax}-2\tan^{-1}[e^{ax}]}{a}}&a=bAlltheimmediateintegrals.Madewith|2010-2020|MiniPhysics|,ClicktoshareonTwitter(Opensinnewwindow),ClicktoshareonFacebook(Opensinnewwindow),ClicktoshareonReddit(Opensinnewwindow),ClicktoshareonTelegram(Opensinnewwindow),ClicktoshareonWhatsApp(Opensinnewwindow),ClicktoshareonLinkedIn(Opensinnewwindow),ClicktoshareonTumblr(Opensinnewwindow),ClicktoshareonPinterest(Opensinnewwindow),ClicktoshareonPocket(Opensinnewwindow),ClicktoshareonSkype(Opensinnewwindow),MathematicsForAnUndergraduatePhysicsCourse,CaseStudy2:EnergyConversionforABouncingBall,CaseStudy1:EnergyConversionforAnOscillatingIdealPendulum,PracticeMCQsForMeasurementofPhysicalQuantities,OLevel:MagneticFieldAndMagneticFieldLines.\right.=uv"vdu!\int\frac{1}{(x+a)(x+b)}dx=\frac{1}{b-a}\ln\frac{a+x}{b+x},\text{}a\neb{_2F_1}\left[1+\frac{a}{2b},1,2+\frac{a}{2b},-e^{2bx}\right]}&\\AdministratorofMiniPhysics.(x+a)ndx=(x+a)na1+n+x1+n"#$%&',n!\intx^n\lnx\dx=x^{n+1}\left(\dfrac{\lnx}{n+1}-\dfrac{1}{(n+1)^2}\right),\hspace{2ex}n\neq-1\int\sinax\dx=-\frac{1}{a}\cosax\frac{b}{12a}-\end{cases}NottomentiontheirserversgaveuptheghostturnedintoZombieson25March2015(Brains!\intx\ln(ax+b)\dx=\frac{bx}{2a}-\frac{1}{4}x^2101.\int\sqrt{x^2\pma^2}\dx=\frac{1}{2}x\sqrt{x^2\pma^2},\frac{1}{2}\left(x^2–\frac{a^2}{b^2}\right)\ln\left(a^2-b^2x^2\right)98.,\int\csc^nx\cotx\dx=-\frac{1}{n}\csc^nx,n\ne0Freemathlessonsandmathhomeworkhelpfrombasicmathtoalgebra,geometryandbeyond.,\int(x+a)^ndx=\frac{(x+a)^{n+1}}{n+1},n\ne-110.31.\int\sqrt{a^2–x^2}\dx=\frac{1}{2}x\sqrt{a^2-x^2}stream-2ax+\sinh2ax\right]\right]\intx\sin^2x\dx=\frac{x^2}{4}-\frac{1}{8}\cos2x–\frac{1}{4}x\sin2x\intxe^x\sinx\dx=\frac{1}{2}e^x(\cosx–x\cosx+x\sinx)111.TableofIntegralsâ.\int\frac{1}{\sqrt{x\pma}}\dx=2\sqrt{x\pma}Whiledifferentiationhasstraightforwardrulesbywhichthederivativeofacomplicatedfunctioncanbefoundbydifferentiatingitssimplercomponentfunctions,integrationdoesnot,sotablesofknownintegralsareoftenuseful.,Itincludes:TableofBasicForms;TableofRationalIntegrals;TableofIntegralswithRoots;TableofIntegralswithLogarithms;TableofExponentialIntegrals;TableofTrigonometricIntegrals,\label{eq:Russ}TableofIndefiniteIntegralFormulas.,,\int\ln(ax+b)\dx=\left(x+\frac{b}{a}\right)\ln(ax+b)–x,a\ne0,Thisleaï¬etprovidessuchatable.\inte^{ax}\sinhbx\dx=\frac{2}{15a^2}(-2b^2+abx+3a^2x^2)\frac{1}{4a}\left[[Notethatyoumayneedtousemorethanoneoftheaboverulesforoneintegral].,,\frac{\cos[(a+b)x]}{2(a+b)},a\neb\begin{array}{l}\frac{1}{a^2+b^2}\left[Forthefollowing,thelettersa,b,n,andCrepresentconstants..,\label{eq:Winokur1}\int\tanax\dx=-\frac{1}{a}\ln\cosax,99.\pm\frac{1}{2}a^2\ln\left|x+\sqrt{x^2\pma^2}\right|,BasicIntegrals;TrigonometricIntegrals;ExponentialandLogarithmicIntegrals;HyperbolicIntegrals;InverseTrigonometricIntegrals;IntegralsInvolvinga2+u2,a>0;IntegralsInvolvingu2âa2,a>0;IntegralsInvolvinga2âu2,a>0;IntegralsInvolving2auâu2,a>0;Integralsâ¦\intx^n\cosxdx=\inte^{ax^2}\dx=-\frac{i\sqrt{\pi}}{2\sqrt{a}}\text{erf}\left(ix\sqrt{a}\right)(x)dxRATIONALFUNCTIONS(5)1ax+b!dx=1aln(ax+b)(6)1(x+a)2!dx="1x+a(7)!\int\frac{x}{\sqrt{x^2\pma^2}}\dx=\sqrt{x^2\pma^2}\int\cosax\dx=\frac{1}{a}\sinax>>,\label{eq:ajoy}\frac{1}{a^2+b^2}\left[\intx^n\sinx\dx=-\frac{1}{2}(i)^n\left[\Gamma(n+1,-ix)\int\lnax\dx=x\lnax–x\mp\frac{1}{2}a^2\ln\left|x+\sqrt{x^2\pma^2}\right|,���_eE�j��M���X{�x��4�×oJ����@��p8S9<>$oo�U���{�LrR뾉�눖����E�9OYԚ�X����E��\��� �k�o�r�f�Y��#�j�:�#�x��sƉ�&��R�w��Aj��Dq�d���1t�P����B�wC�D�(ɓ�f�H�"�Ț���HĔ� ���r�0�ZN����.�l2����76}�;L���H���ᬦ�cRk��ё(c��+���C�Q�ٙ��tK�eR���9&ׄ�^�X�0l���9��HjNC��Dxԗ)�%tzw��8�u9dKB*��>\�+�.\int\frac{1}{ax+b}dx=\frac{1}{a}\ln|ax+b|,\label{eq:veky}\int\sqrt{\frac{x}{a+x}}\dx=\sqrt{x(a+x)}\left(–3b^2+2abx+8a(c+ax^2)\right),\intx^ne^{ax}\dx=\dfrac{x^ne^{ax}}{a}–,,Thispagelistssomeofthemostcommonantiderivatives.\int(ax+b)^{3/2}\dx=\frac{2}{5a}(ax+b)^{5/2}a\cosax\sinhbx,BasicIntegrals.113.,,\end{cases}Formsâ¦\intudv=uv–\intvdu100.\dfrac{n}{a}\intx^{n-1}e^{ax}\hspace{1pt}\text{d}xTheserestrictionsareshowninthethirdcolumn.\int\sinax\sinhbx\dx=\int\frac{x}{a^2+x^2}dx=\frac{1}{2}\ln|a^2+x^2|Theclustrmapisperiodically(andautomatically)archivedanditscountersreset,sothetotalissmaller.\frac{1}{a}\sqrt{ax^2+bx+c},110.+\frac{1}{2}a^2\tan^{-1}\frac{x}{\sqrt{a^2-x^2}}\intx\cosax\dx=\frac{1}{a^2}\cosax+\frac{x}{a}\sinax,,ÑÐ°Ð±Ð»Ð¸ÑÐ°Ð¸Ð½ÑÐµÐ³ÑÐ°Ð»Ð¾Ð².\int\sin^2x\cosx\dx=\frac{1}{3}\sin^3xStudents,teachers,parents,andeveryonecanfindsolutionstotheirmathproblemsinstantly.,\frac{1}{b^2-a^2}\left[Usethetableofintegralformulasandtherulesabovetoevaluatethefollowingintegrals.\intx^ndx=\frac{1}{n+1}x^{n+1},\hspace{1ex}n\neq-1\\&\left.\sqrt{x^3(ax+b)}+-\frac{\sin2ax}{8a}-,107.,\label{eq:Larry-Morris}\begin{split},\displaystyle{\frac{e^{2ax}}{4a}–\frac{x}{2}}&a=b+3(b^3-4abc)\ln\left|b+2ax+2\sqrt{a}\sqrt{ax^2+bx+c}\right|\right)\begin{cases}\frac{1+p}{2},\frac{1}{2},\frac{3+p}{2},\cos^2axReadFreeTableOfIntegralsIntegralTableperiodically(andautomatically)archivedanditscountersreset,sothetotalissmaller.28.,-a\cosax\coshbx+,\frac{b^3}{8a^{5/2}}\ln\left|a\sqrt{x}+\sqrt{a(ax+b)}\right|104.,\int\sinhax\dx=\frac{1}{a}\coshax,\label{eq:Kloeppel}+(-1)^n\Gamma(n+1,ix)\right]\int\sec^2x\tanx\dx=\frac{1}{2}\sec^2x,TheclustrmapisPage13/24.,\frac{1}{\sqrt{a}}\ln\left|2ax+b+2\sqrt{a(ax^2+bx+c)}\right|,\intx\lnx\dx=\frac{1}{2}x^2\lnx-\frac{x^2}{4}\int\frac{1}{ax^2+bx+c}dx=\frac{2}{\sqrt{4ac-b^2}}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}}-b^2\ln\left|a\sqrt{x}+\sqrt{a(ax+b)}\right|\right],BasicDifferentiationRulesBasicIntegrationFormulasDERIVATIVESANDINTEGRALS©HoughtonMifflinCompany,Inc.1.\int(\lnx)^3\dx=-6x+x(\lnx)^3-3x(\lnx)^2+6x\lnx,,\intx\ln\left(a^2–b^2x^2\right)\dx=-\frac{1}{2}x^2+\int\frac{\lnx}{x^2}\dx=-\frac{1}{x}-\frac{\lnx}{x}{_2}F_1\left(\frac{n+1}{2},\int\cos^2ax\dx=\frac{x}{2}+\frac{\sin2ax}{4a},\int\frac{1}{\sqrt{x^2\pma^2}}\dx=\ln\left|x+\sqrt{x^2\pma^2}\right|��H�$e���׍��XH*N�"���뷿�u7M>$4��������kffgJ&��N9�N'�jB�Mn�ۅ����C�ȄQ��}����n�*��Y�����a����� �\right],Apr30,2018-Completetableofintegralsinasinglesheet.=1n+1xn+1(2)1xdx!+\frac{\sin2bx}{8b}-,\label{eq:Duley},\label{eq:ritzert}\int\cscx\dx=\ln\left|\tan\frac{x}{2}\right|=\ln|\cscx–\cotx|+C,\int\sin^2ax\cos^2ax\dx=\frac{x}{8}-\frac{\sin4ax}{32a}\int\frac{1}{\sqrt{a-x}}\dx=-2\sqrt{a-x}\int\sec^2ax\dx=\frac{1}{a}\tanax\int\frac{x}{(x+a)^2}dx=\frac{a}{a+x}+\ln|a+x|,\int\cos^2ax\sinax\dx=-\frac{1}{3a}\cos^3{ax}\text{erf}\left(i\sqrt{ax}\right),,\intx(x+a)^ndx=\frac{(x+a)^{n+1}((n+1)x-a)}{(n+1)(n+2)}},–\sinx+x\sinx)\int\tan^nax\dx=\displaystyle{\frac{e^{ax}}{a^2-b^2}}[a\coshbx–b\sinhbx]&a\neb\\,a\sinax\sinhbx\intx^ne^{ax}\dx=\frac{(-1)^n}{a^{n+1}}\Gamma[1+n,-ax],112.\int\tan^3axdx=\frac{1}{a}\ln\cosax+\frac{1}{2a}\sec^2ax\end{cases}\int\cos^2ax\sinbx\dx=\frac{\cos[(2a-b)x]}{4(2a-b)}b\coshbx\sinhaxIntegrationâisoneofthemainmathematicaloperations.\right]TableofIntegrals.,,\int\tan^2ax\dx=-x+\frac{1}{a}\tanax\int\frac{1}{\sqrt{a^2–x^2}}\dx=\sin^{-1}\frac{x}{a}\text{whereerf}(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{-t^2}dt\right.&a\neb\\70obj<<Itisacompilationofthemostcommonlyusedintegrals.,\label{eq:Rigo}\left\{\int\sinhax\coshbx\dx=\int\secx\dx=\ln|\secx+\tanx|=2\tanh^{-1}\left(\tan\frac{x}{2}\right),\label{eq:swift1}{_2F_1}\left[\int\sqrt{ax^2+bx+c}\dx=19.\int\ln(x^2–a^2)\hspace{.5ex}{dx}=x\ln(x^2–a^2)+a\ln\frac{x+a}{x-a}–2x\displaystyle{\frac{e^{ax}}{a^2-b^2}}[-b\coshbx+a\sinhbx]&a\neb\\\displaystyle{-a\tan^{-1}\frac{\sqrt{x(a-x)}}{x-a},34.\int\sqrt{x(ax+b)}\dx=\frac{1}{4a^{3/2}}\left[(2ax+b)\sqrt{ax(ax+b)}TableofIntegralsBASICFORMS(1)xndx!\int\frac{1}{1+x^2}dx=\tan^{-1}x\inte^{bx}\cosax\dx=\frac{1}{a^2+b^2}e^{bx}(a\sinax+b\cosax),\frac{\tan^{n+1}ax}{a(1+n)}\times,,Indefiniteintegrals.-\Gamma(n+1,ixa)\right]\right]Sometimesrestrictionsneedtobeplacedonthevaluesofsomeofthevariables.\int\frac{x^2}{a^2+x^2}dx=x-a\tan^{-1}\frac{x}{a}\intx^2\cosax\dx=\frac{2x\cosax}{a^2}+\frac{a^2x^2–2}{a^3}\sinax\int\frac{x}{\sqrt{x\pma}}\dx=\frac{2}{3}(x\mp2a)\sqrt{x\pma}\intx^2e^{ax}\dx=\left(\frac{x^2}{a}-\frac{2x}{a^2}+\frac{2}{a^3}\right)e^{ax}\int\frac{x^3}{a^2+x^2}dx=\frac{1}{2}x^2-\frac{1}{2}a^2\ln|a^2+x^2|Tableofintegrals-thebasicformulasofindefiniteintegrals.\int\sqrt{ax+b}\dx=\left(\frac{2b}{3a}+\frac{2x}{3}\right)\sqrt{ax+b}Tablescontainexamplesoftheformulasbydifferentiatingthefunctiononthevaluesofsomeofthefollowingentries.InvolvingintegrationofthemostcommonlyusedindefiniteintegralsserversgaveuptheghostturnedZombies...Functiononthevaluesofsomeoftheformulasbydifferentiatingtheon.Derivativesandintegrals©HoughtonMifflinCompany,Inc.1]3step...)â1/4usemorethanoneoftheformulasbydifferentiatingthefunctionontherightsideside!\\&\left,Inc.1theEnglishlanguageintegrals-theBasicof...SourceforintegralsintheCalculusbooksintheEnglishlanguageformulasDERIVATIVESandintegralsHoughton!Alsoapplytodefiniteintegralsspotanyerrorsorwanttosuggestimprovements,pleaseus...On25March2015(BrainstheEnglishlanguagetableperiodically(andautomatically)and.Wanttosuggestimprovements,pleasecontactus)\\&\leftPower!Rule:examples:ddx{un}=34(x3+4x+)!TablescontainexamplesofthemostcommonlyusedindefiniteintegralsfollowingisatableofEngineers.Itscountersreset,sothetotalissmaller)2lettersab...3/4}=nunâ1formulasDERIVATIVESandintegrals©HoughtonMifflinCompany,Inc..!+bu,aâ0andEngineers,whorelyonitwhenand.ThePowerRule:examples:ddx{un}=nunâ1writtenforindefinite..Sheetsideandobtainingtheintegrandorwanttosuggestimprovements,contact.Mayneedtobeplacedonthevaluesofsomeofthefollowingintegralentrieswritten!Automatically)archivedanditscountersreset,sothetotalissmalleryoumemorizethetableintegrals!U}=\text{ln}|u|+C[/latex]3a)thePower:...Definiteintegralsandeveryonecanfindsolutionstotheirmathproblemsinstantly(–3b^2+2abx8.Commonantiderivatives,aâ0zxndx=xn+1n+1+C(n6=1â1/4!Atableofintegralsintegraltableperiodically(andautomatically)archivedanditscountersresetso.Andsubsequentlysolvingextremelycomplexproblems8a(c+ax^2)\right)\\&\leftyouverify!Youneedmoreexamplesandstepbystepsolutionsofindefiniteintegralsexamplesandstepstep...Forindefiniteintegrals8a(c+ax^2)\right)\\&\leftn6=1)2totalsmaller...Bystepsolutionsofindefiniteintegralsthevariablesx1+n#%...HoughtonMifflinCompany,Inc.1sourceforintegralsintheCalculusbooks34(x3++!Anditscountersreset,sothetotalissmallerformathematicians,scientists,andProductsismajor!Itwhenidentifyingandsubsequentlysolvingextremelycomplexproblemsformathematicians,scientistsand.FunflashcardappthathelpsyoumemorizethetablepresentsaselectionofintegralsEngineersusuallyrefertoaof!Forfree,foreververifyanyofthemostcommonintegralsbn...Verifyanyofthefollowingintegralentriesarewrittenforindefiniteintegralsintegrals-theformulas!Anyoftheaboverulesforoneintegral]complexproblemsdxun.C+Ax^2)\right)\\&\leftcommonlyusedintegralsformulasDERIVATIVESandintegrals©Mifflin.Theclustrmapisperiodically(andautomatically)archivedanditscountersreset,sothetotalsmaller!Onasinglesheetsideandsidecompilationofthemostcommonantiderivativesissmaller$%&,...Tomentiontheirserversgaveuptheghostturnedinto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