samaakalan

kisi falan ka nishchit samaakal (definite integral) us falan ke graaf se ghire kshetr ka chihnasahit kshetrafal dvaara niroopit kiya ja sakta hai. dhaaka

samaakalan (Integral Calculus) yeh ek vishesh prakaar ki yog kriya hai jismein atyanu (infinitesimal) maan waali kintu ginti mein atyadhik char raashiyon ko joda jaata hai. iska ek pramukh upayog vakraakaar kshetron ka kshetrafal nikaalne mein hota hai. samaakalan ko avakalan ki vyutkram sankriya ki tarah bhi samjha ja sakta hai.

falan ${\displaystyle f(x)}$ ka anishchit samaakalan vah falan hai jo nimnalikhit prakaar se paribhaashit hai-

${\displaystyle \int {f(x)}dx=F(x)+c}$

jahaaain, ${\displaystyle c}$ koi niytaank hai ; ${\displaystyle F(x)}$ falan ${\displaystyle f(x)}$ ka samaakalan ya enti-derivative hai ; arthaat ${\displaystyle F'(x)=f(x)}$

${\displaystyle \int f(x)dx}$ ko '${\displaystyle x}$ ke saapeksh ${\displaystyle f(x)}$ ka samaakal' padhte hain.

• ${\displaystyle \int cf(x)\,dx=c\int f(x)\,dx}$
• ${\displaystyle \int [f(x)+g(x)]\,dx=\int f(x)\,dx+\int g(x)\,dx}$
• ${\displaystyle \int f'(x)g(x)\,dx=f(x)g(x)-\int f(x)g'(x)\,dx}$

seedhi vidhi

char parivartan karke samaakalan karna

${\displaystyle \int _{\varphi (a)}^{\varphi (b)}f(x)\,\mathrm {d} x=\int _{a}^{b}f(\varphi (t))\varphi '(t)\,\mathrm {d} t.}$

khandash: samaakalan (inteegresan bye paarts)

${\displaystyle \int _{a}^{b}u(x)v'(x)\,\mathrm {d} x=[u(x)v(x)]_{a}^{b}-\int _{a}^{b}u'(x)v(x)\,\mathrm {d} x.}$

aanshik bhinn vidhi

${\displaystyle \int f(x)dx}$ – anishchit samaakal

${\displaystyle \int \limits _{a}^{b}f(x)dx}$ – nishchit samaakal (Definite integral)

${\displaystyle \int \limits _{-\infty }^{0}f(x)dx}$ – anant samaakal improper integral (=infinite integral)

${\displaystyle \int \limits _{E}f(x)dx}$ – lebeg samaakal (Lebesgue integral)

${\displaystyle \iint \limits _{S}f(x,y,z)\;dS}$ – prushth samaakal (surface integral)

${\displaystyle \oint \limits _{S}f(x,y)\;dl}$ – kisi band vakr ke saapeksh vakrarekhi samaakal

• samaakal soochi (Table of Integrals)
• avakalan (Differentiation)

• Riemann Sum by Wolfram Research
• Introduction to definite integrals by Khan Academy

aanalaain pustakein

• Keisler, H. Jerome, Elementary Calculus: An Approach Using Infinitesimals, University of Wisconsin
• Stroyan, K.D., A Brief Introduction to Infinitesimal Calculus, University of Iowa
• Mauch, Sean, Sean's Applied Math Book, CIT, an online textbook that includes a complete introduction to calculus
• Crowell, Benjamin, Calculus, Fullerton College, an online textbook
• Garrett, Paul, Notes on First-Year Calculus
• Hussain, Faraz, Understanding Calculus, an online textbook
• Kowalk, W.P., Integration Theory, University of Oldenburg. A new concept to an old problem. Online textbook
• Sloughter, Dan, Difference Equations to Differential Equations, an introduction to calculus
• Numerical Methods of Integration at Holistic Numerical Methods Institute
• P.S. Wang, Evaluation of Definite Integrals by Symbolic Manipulation (1972) - a cookbook of definite integral techniques