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\int e^{{cx}}\;{\mathrm {d}}x={\frac {1}{c}}e^{{cx}}

\int a^{{cx}}\;{\mathrm {d}}x={\frac {1}{c\cdot \ln a}}a^{{cx}} for a>0,\ a\neq 1

\int xe^{{cx}}\;{\mathrm {d}}x={\frac {e^{{cx}}}{c^{2}}}(cx-1)

\int x^{2}e^{{cx}}\;{\mathrm {d}}x=e^{{cx}}\left({\frac {x^{2}}{c}}-{\frac {2x}{c^{2}}}+{\frac {2}{c^{3}}}\right)

\int x^{n}e^{{cx}}\;{\mathrm {d}}x={\frac {1}{c}}x^{n}e^{{cx}}-{\frac {n}{c}}\int x^{{n-1}}e^{{cx}}{\mathrm {d}}x=\left({\frac {\partial }{\partial c}}\right)^{n}{\frac {e^{{cx}}}{c}}

\int {\frac {e^{{cx}}}{x}}\;{\mathrm {d}}x=\ln |x|+\sum _{{n=1}}^{\infty }{\frac {(cx)^{n}}{n\cdot n!}}

\int {\frac {e^{{cx}}}{x^{n}}}\;{\mathrm {d}}x={\frac {1}{n-1}}\left(-{\frac {e^{{cx}}}{x^{{n-1}}}}+c\int {\frac {e^{{cx}}}{x^{{n-1}}}}\,{\mathrm {d}}x\right)\qquad {\mbox{(for }}n\neq 1{\mbox{)}}

\int e^{{cx}}\ln x\;{\mathrm {d}}x={\frac {1}{c}}\left(e^{{cx}}\ln |x|-\operatorname {Ei}\,(cx)\right)

\int e^{{cx}}\sin bx\;{\mathrm {d}}x={\frac {e^{{cx}}}{c^{2}+b^{2}}}(c\sin bx-b\cos bx)

\int e^{{cx}}\cos bx\;{\mathrm {d}}x={\frac {e^{{cx}}}{c^{2}+b^{2}}}(c\cos bx+b\sin bx)

\int e^{{cx}}\sin ^{n}x\;{\mathrm {d}}x={\frac {e^{{cx}}\sin ^{{n-1}}x}{c^{2}+n^{2}}}(c\sin x-n\cos x)+{\frac {n(n-1)}{c^{2}+n^{2}}}\int e^{{cx}}\sin ^{{n-2}}x\;{\mathrm {d}}x

\int e^{{cx}}\cos ^{n}x\;{\mathrm {d}}x={\frac {e^{{cx}}\cos ^{{n-1}}x}{c^{2}+n^{2}}}(c\cos x+n\sin x)+{\frac {n(n-1)}{c^{2}+n^{2}}}\int e^{{cx}}\cos ^{{n-2}}x\;{\mathrm {d}}x

\int xe^{{cx^{2}}}\;{\mathrm {d}}x={\frac {1}{2c}}\;e^{{cx^{2}}}

\int e^{{-cx^{2}}}\;{\mathrm {d}}x={\sqrt {{\frac {\pi }{4c}}}}\operatorname {erf}({\sqrt {c}}x) (\operatorname {erf} is the error function)

\int xe^{{-cx^{2}}}\;{\mathrm {d}}x=-{\frac {1}{2c}}e^{{-cx^{2}}}

\int {\frac {e^{{-x^{2}}}}{x^{2}}}\;{\mathrm {d}}x=-{\frac

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... | | |leaving the final simplified form of | | | | |the derivative | Set[pic]and solve for x to find critical point(s): When the derivative is set to zero, [pic]; thus, this implies each factor could be equal to zero, meaning that there could be up to three values for x. |[pic] |= |[pic] |set first factor equal to zero | |[pic] |= |[pic] |divide each side of the equation by 2| |[pic] |= |[pic]...

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...Project Gutenberg EBook of Calculus Made Easy, by Silvanus Thompson This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: Calculus Made Easy Being a very-simplest introduction to those beautiful methods which are generally called by the terrifying names of the Differentia Author: Silvanus Thompson Release Date: October 9, 2012 [EBook #33283] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK CALCULUS MADE EASY *** Produced by Andrew D. Hwang, Brenda Lewis and the Online Distributed Proofreading Team at http://www.pgdp.net (This file was produced from images generously made available by The Internet Archive/American Libraries.) transcriber’s note Minor presentational changes, and minor typographical and numerical corrections, have been made without comment. All A textual changes are detailed in the L TEX source ﬁle. This PDF ﬁle is optimized for screen viewing, but may easily be A recompiled for printing. Please see the preamble of the L TEX source ﬁle for instructions. CALCULUS MADE EASY MACMILLAN AND CO., Limited LONDON : BOMBAY : CALCUTTA MELBOURNE THE MACMILLAN COMPANY NEW YORK : BOSTON : CHICAGO DALLAS : SAN FRANCISCO THE MACMILLAN CO. OF CANADA, Ltd. TORONTO CALCULUS MADE EASY: BEING A VERY-SIMPLEST...

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...Calculus From Wikipedia, the free encyclopedia This article is about the branch of mathematics. For other uses, see Calculus (disambiguation). Topics in Calculus Fundamental theorem Limits of functions Continuity Mean value theorem [show]Differential calculus [show]Integral calculus [show]Vector calculus [show]Multivariable calculus Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics focused on limits,functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modernmathematics education. It has two major branches,differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change,[1] in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has widespread applications in science,economics, and engineering and can solve many problems for which algebra alone is insufficient. Historically, calculus was called "the calculus of infinitesimals", or "infinitesimal calculus". More generally, calculus (plural calculi) refers to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus...

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