# table of basic integrals

\text{erf}\left(i\sqrt{ax}\right),,,,\label{eq:xul},,,,,ÑÐ°Ð±Ð»Ð¸ÑÐ°Ð¸Ð½ÑÐµÐ³ÑÐ°Ð»Ð¾Ð².\intxe^x\cosx\dx=\frac{1}{2}e^x(x\cosx\int\sin^2ax\cos^2bxdx=\frac{x}{4}\frac{1}{a^2+b^2}\left[-a\cosax\coshbx+,\frac{b}{12a}-TableofBasicIntegrals1(1)Zxndx=1n+1xn+1;n6=1(2)Z1xdx=lnjxj(3)Zudv=uvZvdu(4)Zexdx=e(5)Zaxdx=1lnaax(6)Zlnxdx=xlnxx(7)Zsinxdx=cosx(8)Zcosxdx=sinx(9)Ztanxdx=lnjsecxj(10)Zsecxdx=lnjsecx+tanxj(11)Zsec2xdx=tanx(12)Zsecxtanxdx=secx(13)Zaa2+x2dx=tan1xa(14)Zaa2x2\end{cases}99.\displaystyle{\frac{e^{2ax}}{4a}+\frac{x}{2}}&a=b\int\frac{x^3}{a^2+x^2}dx=\frac{1}{2}x^2-\frac{1}{2}a^2\ln|a^2+x^2|\right]\intx^2\sinax\dx=\frac{2-a^2x^2}{a^3}\cosax+\frac{2x\sinax}{a^2}\int\sin^2ax\dx=\frac{x}{2}–\frac{\sin2ax}{4a}\right.Apr30,2018-Completetableofintegralsinasinglesheet.Formscontainingtrigonometricfunctions.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!\end{cases}Tableofintegrals-thebasicformulasofindefiniteintegrals.Formulas:-BasicIntegrationFormulas-Integralsoftherationalfunctionsofpart-Integralsoftranscendentalfunctions-Integralsoftheirrationalfunctionsofpart-Integralsoftrigonometricfunctionsofpart-Propertyofindeterminateintegrals-PropertiesoftheDefiniteIntegral\int\frac{1}{\sqrt{ax^2+bx+c}}\dx=\intx^n\sinx\dx=-\frac{1}{2}(i)^n\left[\Gamma(n+1,-ix),\label{eq:Gilmore}\intx^n\cosxdx=\frac{\sin[2(a+b)x]}{16(a+b)}\begin{cases},&a\neb\\Formulas:-BasicIntegrationFormulas-Integralsoftherationalfunctionsofpart-Integralsoftranscendentalfunctions-Integralsoftheirrationalfunctionsofpart-Integralsoftrigonometricfunctionsofpart-Propertyofindeterminateintegrals-PropertiesoftheDefiniteIntegral\int\secx\cscx\dx=\ln|\tanx|,,\label{eq:veky},\inte^{ax}\coshbx\dx=\int\tanax\dx=-\frac{1}{a}\ln\cosax–\sinx+x\sinx)Therehavebeenvisitorstointegral-table.comsince2004.,\int\frac{\lnax}{x}\dx=\frac{1}{2}\left(\lnax\right)^2\int\secx\tanx\dx=\secx\int\ln(x^2–a^2)\hspace{.5ex}{dx}=x\ln(x^2–a^2)+a\ln\frac{x+a}{x-a}–2x\frac{1}{2}\left(x^2–\frac{a^2}{b^2}\right)\ln\left(a^2-b^2x^2\right)\frac{1}{2}(ia)^{1-n}\left[(-1)^n\Gamma(n+1,-iax),\intx\cos^2x\dx=\frac{x^2}{4}+\frac{1}{8}\cos2x+\frac{1}{4}x\sin2x\intx(x+a)^ndx=\frac{(x+a)^{n+1}((n+1)x-a)}{(n+1)(n+2)}\inte^{bx}\cosax\dx=\frac{1}{a^2+b^2}e^{bx}(a\sinax+b\cosax)\intx\sqrt{x^2\pma^2}\dx=\frac{1}{3}\left(x^2\pma^2\right)^{3/2}TableofIntegralsEngineersusuallyrefertoatableofintegralswhenperformingcalculationsinvolvingintegration.\int\frac{x}{\sqrt{ax^2+bx+c}}\dx=,4.,\label{eq:Winokur1}FreeIntegrationWorksheet.IntegralsInvolvinga+bu,aâ 0.,\\\frac{2}{3}x(x-a)^{3/2}–\frac{4}{15}(x-a)^{5/2},\text{or}\int\frac{1}{1+x^2}dx=\tan^{-1}x\int\tan^3axdx=\frac{1}{a}\ln\cosax+\frac{1}{2a}\sec^2ax\int\sinhax\dx=\frac{1}{a}\coshax,\int\cos^3axdx=\frac{3\sinax}{4a}+\frac{\sin3ax}{12a}22.,TheclustrmapisPage13/24.TableofIntegralsâ.\inte^{ax^2}\dx=-\frac{i\sqrt{\pi}}{2\sqrt{a}}\text{erf}\left(ix\sqrt{a}\right)\displaystyle{\frac{e^{2ax}}{4a}–\frac{x}{2}}&a=b-2x+\left(\frac{b}{2a}+x\right)\ln\left(ax^2+bx+c\right)Formsâ¦\int(x+a)^ndx=\frac{(x+a)^{n+1}}{n+1},n\ne-1112.NottomentiontheirserversA:TABLEOFBASICDERIVATIVESLetu=u(x)beadifferentiablefunctionoftheindependentvariablex,thatisu(x)exists.70obj<<–\frac{\sin[(2a+b)x]}{4(2a+b)}{_2F_1}\left[1+\frac{a}{2b},1,2+\frac{a}{2b},-e^{2bx}\right]}&\\,,,$$\int\limits^{+\infty}_{-\infty}e^{-ax^{2}}=\sqrt{\frac{\pi}{a}}$$,$$\int\limits^{+\infty}_{-\infty}x^{2n}e^{-ax^{2}}=(-1)^{n}\frac{\partial^{n}}{\partiala^{n}}\sqrt{\frac{\pi}{a}}$$,$$\int\limits^{+\infty}_{-\infty}e^{-ax^{2}+bx}=e^{\frac{b^2}{4a}}\sqrt{\frac{\pi}{a}}$$,$$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}}x^{2}\sin^2\left(\frac{n\pix}{a}\right)=\frac{1}{24}a^{3}\left(1–\frac{6(-1)^n}{n^2\pi^2}\right)$$,$$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}}x^{2}\cos^2\left(\frac{n\pix}{a}\right)=\frac{1}{24}a^{3}\left(1+\frac{6(-1)^n}{n^2\pi^2}\right)$$,$$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}}xÂ \cos\left(\frac{\pix}{a}\right)Â \sin\left(\frac{2\pix}{a}\right)=\frac{8a^2}{9\pi^2}$$,$$\int\limits^{a}_{b}\frac{dx}{\sqrt{\left(a-x\right)\left(x-b\right)}}=\pi\text{fora>b}$$,$$\int\limits^{a}_{b}\frac{dx}{x\sqrt{\left(a-x\right)\left(x-b\right)}}=\frac{\pi}{\sqrt{ab}}\text{fora>b>0}$$,$$\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\frac{dx}{1+y\sinx}=\frac{\pi}{\sqrt{1–y^2}}\text{for-1<y<1}$$,$$\int\frac{dx}{\sqrt{a^{2}–x^{2}}}=\text{arcsin}\,\frac{x}{a}$$,$$\int\frac{xdx}{\sqrt{a^{2}+x^{2}}}=\sqrt{a^{2}+x^{2}}$$,$$\int\frac{dx}{\sqrt{a^{2}+x^{2}}}=\text{ln}\,\left(x+\sqrt{a^{2}+x^{2}}\right)$$,$$\int\frac{dx}{a^{2}+x^{2}}=\frac{1}{a}\,\text{arctan}\,\frac{x}{a}$$,$$\int\frac{dx}{\left(a^{2}+x^{2}\right)^{\frac{3}{2}}}=\frac{1}{a^{2}}\frac{x}{\sqrt{a^{2}+x^{2}}}$$,$$\int\frac{x\,dx}{\left(a^{2}+x^{2}\right)^{\frac{3}{2}}}=\,–\frac{1}{\sqrt{a^{2}+x^{2}}}$$,$$\int\frac{dx}{\sqrt{(x–a)^{2}+b^{2}}}=\text{ln}\,\frac{1}{(a–x)+\sqrt{(a-x)^{2}+b^{2}}}$$,$$\int\frac{(x–a)\,dx}{\left[(x-a)^{2}+b^{2}\right]^{\frac{3}{2}}}=\,–\frac{1}{\sqrt{(x-a)^{2}+b^{2}}}$$,$$\int\frac{dx}{\left[(x–a)^{2}+b^{2}\right]^{\frac{3}{2}}}=\frac{x–a}{b^{2}\sqrt{(x–a)^{2}+b^{2}}}$$.>>\int\cos^2ax\dx=\frac{x}{2}+\frac{\sin2ax}{4a}���_eE�j��M���X{�x��4�×oJ����@��p8S9<>$oo�U���{�LrR뾉�눖����E�9OYԚ�X����E��\��� �k�o�r�f�Y��#�j�:�#�x��sƉ�&��R�w��Aj��Dq�d���1t�P����B�wC�D�(ɓ�f�H�"�Ț���HĔ� ���r�0�ZN����.�l2����76}�;L���H���ᬦ�cRk��ё(c��+���C�Q�ٙ��tK�eR���9&ׄ�^�X�0l���9��HjNC��Dxԗ)�%tzw��8�u9dKB*��>\�+�.107.\int\sqrt{a^2–x^2}\dx=\frac{1}{2}x\sqrt{a^2-x^2},,\\&\left.,-\frac{\sin[(2a-b)x]}{4(2a-b)}��H�$e���׍��XH*N�"���뷿�u7M>$4��������kffgJ&��N9�N'�jB�Mn�ۅ����C�ȄQ��}����n�*��Y�����a����� 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