সমাকলনের তালিকাসমূহ

সমাকলন বা যোগজীকরণ (Integration) হচ্ছে সমাকলনবিদ্যার (integral calculus) মৌলিক অংশ। অবকলন (differentiation) এর কতগুলো খুব সহজ সূত্র রয়েছে, যার মাধ্যমে অবকলন করে জটিল ফাংশনগুলোর সাধিত পদ খুব সহজে বের করা যায়, যা সমাকলন বা যোগজীকরণ (Integration)-এ হয় না। এজন্য কিছু পরিচিত রাশি ব্যবহার হয়ে থাকে। এখানে কিছু বহুল প্রচলিত অনির্দিষ্ট সমাকল (ইংরেজি: indefinite integral) এর তালিকা তুলে ধরা হল; ∫ d x = x + C {\displaystyle \int \,{\rm {d}}x=x+C} ∫ x n d x = x n + 1 n + 1 + C  if  n ≠ − 1 {\displaystyle \int x^{n}\,{\rm {d}}x={\frac {x^{n+1}}{n+1}}+C\qquad {\mbox{ if }}n\neq -1} ∫ e x d x = e x + C {\displaystyle \int e^{x}\,dx=e^{x}+C} ∫ a x d x = a x ln ⁡ a + C {\displaystyle \int a^{x}\,dx={\frac {a^{x}}{\ln {a}}}+C} ∫ d x x = ln ⁡ | x | + C {\displaystyle \int {dx \over x}=\ln {\left|x\right|}+C} ∫ ln ⁡ x d x = x ln ⁡ x − x + C {\displaystyle \int \ln {x}\,dx=x\ln {x}-x+C} ∫ sin ⁡ x d x = − cos ⁡ x + C {\displaystyle \int \sin {x}\,dx=-\cos {x}+C} ∫ cos ⁡ x d x = sin ⁡ x + C {\displaystyle \int \cos {x}\,dx=\sin {x}+C} ∫ tan ⁡ x d x = ln ⁡ | sec ⁡ x | + C {\displaystyle \int \tan {x}\,dx=\ln {\left|\sec {x}\right|}+C} ∫ cot ⁡ x d x = ln ⁡ | sin ⁡ x | + C {\displaystyle \int \cot {x}\,dx=\ln {\left|\sin {x}\right|}+C} ∫ sec ⁡ x d x = ln ⁡ | sec ⁡ x + tan ⁡ x | + C {\displaystyle \int \sec {x}\,dx=\ln {\left|\sec {x}+\tan {x}\right|}+C} ∫ cos ⁡ x d x = ln ⁡ | cos ⁡ x − cot ⁡ x | + C {\textstyle \int \cos {x}\,dx=\ln {\left|\cos {x}-\cot {x}\right|}+C} ∫ sec 2 ⁡ x d x = tan ⁡ x + C {\displaystyle \int \sec ^{2}x\,dx=\tan x+C} ∫ csc 2 ⁡ x d x = − cot ⁡ x + C {\displaystyle \int \csc ^{2}x\,dx=-\cot x+C} ∫ sec ⁡ x tan ⁡ x d x = sec ⁡ x + C {\displaystyle \int \sec {x}\,\tan {x}\,dx=\sec {x}+C} ∫ csc ⁡ x cot ⁡ x d x = − csc ⁡ x + C {\displaystyle \int \csc {x}\,\cot {x}\,dx=-\csc {x}+C} sin n ⁡ x {\displaystyle \sin ^{n}x} ইত্যাদি কয়েকটি সমাকল সূত্র ∫ sin n ⁡ x d x = − sin n − 1 ⁡ x cos ⁡ x n + n − 1 n ∫ sin n − 2 ⁡ x d x {\displaystyle \int \sin ^{n}x\,dx=-{\frac {\sin ^{n-1}{x}\cos {x}}{n}}+{\frac {n-1}{n}}\int \sin ^{n-2}{x}\,dx} ∫ cos n ⁡ x d x = cos n − 1 ⁡ x sin ⁡ x n + n − 1 n ∫ cos n − 2 ⁡ x d x {\displaystyle \int \cos ^{n}x\,dx={\frac {\cos ^{n-1}{x}\sin {x}}{n}}+{\frac {n-1}{n}}\int \cos ^{n-2}{x}\,dx} ∫ d x a 2 − x 2 = sin − 1 ⁡ x a + C {\displaystyle \int {dx \over {\sqrt {a^{2}-x^{2}}}}=\sin ^{-1}{x \over a}+C} ∫ − d x a 2 − x 2 = cos − 1 ⁡ x a + C {\displaystyle \int {-dx \over {\sqrt {a^{2}-x^{2}}}}=\cos ^{-1}{x \over a}+C} ∫ d x x x 2 − a 2 = 1 a sec − 1 ⁡ | x | a + C {\displaystyle \int {dx \over x{\sqrt {x^{2}-a^{2}}}}={1 \over a}\sec ^{-1}{|x| \over a}+C} ∫ sinh ⁡ x d x = cosh ⁡ x + C {\displaystyle \int \sinh x\,dx=\cosh x+C} ∫ cosh ⁡ x d x = sinh ⁡ x + C {\displaystyle \int \cosh x\,dx=\sinh x+C} ∫ tanh ⁡ x d x = ln ⁡ | cosh ⁡ x | + C {\displaystyle \int \tanh x\,dx=\ln |\cosh x|+C} ∫ csch x d x = ln ⁡ | tanh ⁡ x 2 | + C {\displaystyle \int {\mbox{csch}}\,x\,dx=\ln \left|\tanh {x \over 2}\right|+C} ∫ sech x d x = arctan ⁡ ( sinh ⁡ x ) + C {\displaystyle \int {\mbox{sech}}\,x\,dx=\arctan(\sinh x)+C} ∫ coth ⁡ x d x = ln ⁡ | sinh ⁡ x | + C {\displaystyle \int \coth x\,dx=\ln |\sinh x|+C} ∫ sech 2 x d x = tanh ⁡ x + C {\displaystyle \int {\mbox{sech}}^{2}x\,dx=\tanh x+C} ∫ arsinh x d x = x arsinh x − x 2 + 1 + C {\displaystyle \int \operatorname {arsinh} \,x\,dx=x\,\operatorname {arsinh} \,x-{\sqrt {x^{2}+1}}+C} ∫ arcosh x d x = x arcosh x − x + 1 x − 1 + C {\displaystyle \int \operatorname {arcosh} \,x\,dx=x\,\operatorname {arcosh} \,x-{\sqrt {x+1}}\,{\sqrt {x-1}}+C} ∫ artanh x d x = x artanh x + ln ⁡ ( x 2 − 1 ) 2 + C {\displaystyle \int \operatorname {artanh} \,x\,dx=x\,\operatorname {artanh} \,x+{\frac {\ln \left(\,x^{2}-1\right)}{2}}+C} ∫ arcoth x d x = x arcoth x + ln ⁡ ( 1 − x 2 ) 2 + C {\displaystyle \int \operatorname {arcoth} \,x\,dx=x\,\operatorname {arcoth} \,x+{\frac {\ln \left(1-x^{2}\right)}{2}}+C} ∫ arsech x d x = x arsech x − 2 arctan ⁡ 1 − x 1 + x + C {\displaystyle \int \operatorname {arsech} \,x\,dx=x\,\operatorname {arsech} \,x-2\,\arctan {\sqrt {\frac {1-x}{1+x}}}+C} ∫ arcsch x d x = x arcsch x + artanh ⁡ 1 x 2 + 1 + C {\displaystyle \int \operatorname {arcsch} \,x\,dx=x\,\operatorname {arcsch} \,x+\operatorname {artanh} {\sqrt {{\frac {1}{x^{2}}}+1}}+C} ∫ cos ⁡ a x e b x d x = e b x a 2 + b 2 ( a sin ⁡ a x + b cos ⁡ a x ) + C {\displaystyle \int \cos ax\,e^{bx}\,dx={\frac {e^{bx}}{a^{2}+b^{2}}}\left(a\sin ax+b\cos ax\right)+C} ∫ sin ⁡ a x e b x d x = e b x a 2 + b 2 ( b sin ⁡ a x − a cos ⁡ a x ) + C {\displaystyle \int \sin ax\,e^{bx}\,dx={\frac {e^{bx}}{a^{2}+b^{2}}}\left(b\sin ax-a\cos ax\right)+C} ∫ cos ⁡ a x cosh ⁡ b x d x = 1 a 2 + b 2 ( a sin ⁡ a x cosh ⁡ b x + b cos ⁡ a x sinh ⁡ b x ) + C {\displaystyle \int \cos ax\,\cosh bx\,dx={\frac {1}{a^{2}+b^{2}}}\left(a\sin ax\,\cosh bx+b\cos ax\,\sinh bx\right)+C} ∫ sin ⁡ a x cosh ⁡ b x d x = 1 a 2 + b 2 ( b sin ⁡ a x sinh ⁡ b x − a cos ⁡ a x cosh ⁡ b x ) + C {\displaystyle \int \sin ax\,\cosh bx\,dx={\frac {1}{a^{2}+b^{2}}}\left(b\sin ax\,\sinh bx-a\cos ax\,\cosh bx\right)+C} ∫ 0 ∞ e − x 2 d x = 1 2 π {\displaystyle \int _{0}^{\infty }{e^{-x^{2}}\,dx}={\frac {1}{2}}{\sqrt {\pi }}} (গাউসের সমাকলন) ∫ 0 ∞ sin ⁡ ( x ) x d x = π 2 {\displaystyle \int _{0}^{\infty }{\frac {\sin(x)}{x}}\,dx={\frac {\pi }{2}}} টেমপ্লেট:AS ref Bronstein, Ilja Nikolaevič; Semendjajew, Konstantin Adolfovič (১৯৮৭) [1945]। Grosche, Günter; Ziegler, Viktor; Ziegler, Dorothea, সম্পাদকগণ। Taschenbuch der Mathematik (German ভাষায়)। 1। Ziegler, Viktor কর্তৃক অনূদিত। Weiß, Jürgen (23 সংস্করণ)। Thun and Frankfurt am Main: Verlag Harri Deutsch (and B. G. Teubner Verlagsgesellschaft, Leipzig)। আইএসবিএন 3-87144-492-8। উদ্ধৃতি শৈলী রক্ষণাবেক্ষণ: অচেনা ভাষা (link) Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (২০১৫) [October 2014]। Zwillinger, Daniel; Moll, Victor Hugo, সম্পাদকগণ। Table of Integrals, Series, and Products (English ভাষায়)। Scripta Technica, Inc. কর্তৃক অনূদিত (8 সংস্করণ)। Academic Press, Inc.। আইএসবিএন 978-0-12-384933-5। এলসিসিএন 2014010276। উদ্ধৃতি শৈলী রক্ষণাবেক্ষণ: অচেনা ভাষা (link) (Several previous editions as well.) Prudnikov, Anatolii Platonovich (Прудников, Анатолий Платонович); Brychkov, Yuri A. (Брычков, Ю. А.); Marichev, Oleg Igorevich (Маричев, Олег Игоревич) (১৯৮৮–১৯৯২) [1981−1986 (Russian)]। Integrals and Series (English ভাষায়)। 1–5। Queen, N. M. কর্তৃক অনূদিত (1 সংস্করণ)। (Nauka) Gordon & Breach Science Publishers/CRC Press। আইএসবিএন 2-88124-097-6। উদ্ধৃতি শৈলী রক্ষণাবেক্ষণ: অচেনা ভাষা (link) . Second revised edition (Russian), volume 1–3, Fiziko-Matematicheskaya Literatura, 2003. Yuri A. Brychkov (Ю. А. Брычков), Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas. Russian edition, Fiziko-Matematicheskaya Literatura, 2006. English edition, Chapman & Hall/CRC Press, 2008, আইএসবিএন ১-৫৮৪৮৮-৯৫৬-X/9781584889564. Daniel Zwillinger. CRC Standard Mathematical Tables and Formulae, 31st edition. Chapman & Hall/CRC Press, 2002. আইএসবিএন ১-৫৮৪৮৮-২৯১-৩. (Many earlier editions as well.) Meyer Hirsch [de], Integraltafeln oder Sammlung von Integralformeln (Duncker und Humblot, Berlin, 1810) Meyer Hirsch [de], Integral Tables Or A Collection of Integral Formulae (Baynes and son, London, 1823) [English translation of Integraltafeln] David Bierens de Haan, Nouvelles Tables d'Intégrales définies (Engels, Leiden, 1862) Benjamin O. Pierce A short table of integrals - revised edition (Ginn & co., Boston, 1899) সমাকলন অন্তরকলন সূচী Paul's Online Math Notes A. Dieckmann, Table of Integrals (Elliptic Functions, Square Roots, Inverse Tangents and More Exotic Functions): Indefinite Integrals Definite Integrals Math Major: A Table of Integrals O'Brien, Francis J. Jr.। "500 Integrals of Elementary and Special Functions"।  Derived integrals of exponential, logarithmic functions and special functions. Rule-based Integration Precisely defined indefinite integration rules covering a wide class of integrands Mathar, Richard J. (২০১২)। "Yet another table of integrals"। arXiv:1207.5845  [math.CA]। উদ্ধৃতি টেমপ্লেট ইংরেজি প্যারামিটার ব্যবহার করেছে (link) Victor Hugo Moll, The Integrals in Gradshteyn and Ryzhik Integration examples for Wolfram Alpha wxmaxima gui for Symbolic and numeric resolution of many mathematical problems ওয়েব্যাক মেশিনে আর্কাইভকৃত ২৭ আগস্ট ২০০৫ তারিখে The Single Most Overpowered Integration Technique in Existence. YouTube Video by Flammable Maths on symmetries গণিত প্রবেশদ্বার