İntegral

İntegral namey leteo ke binê fonksiyoni de ca geno. Seba fonksiyonê deriativ vıraştış asêniye dano.

Formulê cı

${\displaystyle F(x)=\int f(x)+c,}$ o. Raya Limiti ra formulê ho u pêrokerdışê Riemanni ;

${\displaystyle S=\lim _{\Delta x\to 0}\sum _{i=0}^{n-1}f(x_{i})\Delta x_{i}=\int _{a}^{b}f(x)\,dx=F(b)-F(a)}$

Metodê integral-gırewtışi

Seba integral grotışi raya tewr asêni varyan vurnayışo; Bin de jü polinome esto u eno polinome 6. derece rao. Eno polinom ebe fonksiyonê basiti ra çozkerdene mumkın niyo. Na riye ra varyant vurnayış seba çozkerdene asêniye dano:

${\displaystyle x^{6}-9x^{3}+8=0.\,}$

Ena denkleme de x³ = u ebe vurnayışê varyanti denlkem bin de ma nuseni ;

${\displaystyle u^{2}-9u+8=0\,}$

Ena raya ra denklem derecey 2. ra biya. Rıstımê enay denkleme ;

${\displaystyle u=1\quad {\mbox{ve}}\quad u=8'dir.}$

Ebe ena vurnayışo newa ra netıcey varyantê esasi de cay ho ra bınusi ;

${\displaystyle x^{3}=1\quad {\mbox{ve}}\quad x^{3}=8\quad \Rightarrow \qquad x=(1)^{1/3}=1\quad {\mbox{ve}}\quad x=(8)^{1/3}=2.\,}$

İntegralê fonksiyonê basiti

Fonksiyonê rasyonali

${\displaystyle \int dx=x+C}$

${\displaystyle \int x^{n}\,{\rm {d}}x={\frac {x^{n+1}}{n+1}}+C\qquad {\mbox{ eğer }}n\neq -1}$

${\displaystyle \int {dx \over x}=\ln {\left|x\right|}+C}$

${\displaystyle \int {dx \over {a^{2}+x^{2}}}={1 \over a}\arctan {x \over a}+C}$

Fonksiyonê irrasyonali

${\displaystyle \int {dx \over {\sqrt {a^{2}-x^{2}}}}=\arcsin {x \over a}+C}$

${\displaystyle \int {-dx \over {\sqrt {a^{2}-x^{2}}}}=\arccos {x \over a}+C}$

${\displaystyle \int {dx \over x{\sqrt {x^{2}-a^{2}}}}={1 \over a}\sec {|x| \over a}+C}$

Fonksiyonê logaritmiki

${\displaystyle \int \ln(x)\,dx=x\ln(x)-x+C,}$

${\displaystyle \int \log _{b}{x}\,dx=x\log _{b}{x}-x\log _{b}{e}+C}$

Fonskyionê ke serê cı esto

${\displaystyle \int e^{x}\,dx=e^{x}+C}$

${\displaystyle \int a^{x}\,dx={\frac {a^{x}}{\ln {a}}}+C}$

${\displaystyle \int a^{ln(x)}\,dx=\int x^{ln(a)}\,dx={\frac {x\,a^{ln(x)}}{\ln {a}+1}}+C={\frac {x\,x^{ln(a)}}{\ln {a}+1}}+C}$

Fonksiyonê trigonometriki

${\displaystyle \int \sin {x}\,dx=-\cos {x}+C}$

${\displaystyle \int \cos {x}\,dx=\sin {x}+C}$

${\displaystyle \int \tan {x}\,dx=-\ln {\left|\cos {x}\right|}+C}$

${\displaystyle \int \cot {x}\,dx=\ln {\left|\sin {x}\right|}+C}$

${\displaystyle \int \sec {x}\,dx=\ln {\left|\sec {x}+\tan {x}\right|}+C}$

${\displaystyle \int \csc {x}\,dx=\ln {\left|\csc {x}-\cot {x}\right|}+C}$

${\displaystyle \int \sec ^{2}x\,dx=\tan x+C}$

${\displaystyle \int \csc ^{2}x\,dx=-\cot x+C}$

${\displaystyle \int \sec {x}\,\tan {x}\,dx=\sec {x}+C}$

${\displaystyle \int \csc {x}\,\cot {x}\,dx=-\csc {x}+C}$

${\displaystyle \int \sin ^{2}x\,dx={\frac {1}{2}}(x-\sin x\cos x)+C}$

${\displaystyle \int \cos ^{2}x\,dx={\frac {1}{2}}(x+\sin x\cos x)+C}$

${\displaystyle \int \sec ^{3}x\,dx={\frac {1}{2}}\sec x\tan x+{\frac {1}{2}}\ln |\sec x+\tan x|+C}$

${\displaystyle \int \sin ^{n}x\,dx=-{\frac {\sin ^{n-1}{x}\cos {x}}{n}}+{\frac {n-1}{n}}\int \sin ^{n-2}{x}\,dx}$

${\displaystyle \int \cos ^{n}x\,dx={\frac {\cos ^{n-1}{x}\sin {x}}{n}}+{\frac {n-1}{n}}\int \cos ^{n-2}{x}\,dx}$

${\displaystyle \int \arctan {x}\,dx=x\,\arctan {x}-{\frac {1}{2}}\ln {\left|1+x^{2}\right|}+C}$

Fonksiyonê hiperboliki

${\displaystyle \int \sinh x\,dx=-\,coshx+C}$

${\displaystyle \int \cosh x\,dx=\sinh x+C}$

${\displaystyle \int \tanh x\,dx=\ln |\cosh x|+C}$

${\displaystyle \int {\mbox{csch}}\,x\,dx=\ln \left|\tanh {x \over 2}\right|+C}$

${\displaystyle \int {\mbox{sech}}\,x\,dx=\arctan(\sinh x)+C}$

${\displaystyle \int \coth x\,dx=\ln |\sinh x|+C}$

${\displaystyle \int {\mbox{sech}}^{2}x\,dx=\tanh x+C}$

Fonksiyonê hiperbolikio ters

${\displaystyle \int \operatorname {arcsinh} x\,dx=x\operatorname {arcsinh} x-{\sqrt {x^{2}+1}}+C}$

${\displaystyle \int \operatorname {arccosh} x\,dx=x\operatorname {arccosh} x-{\sqrt {x^{2}-1}}+C}$

${\displaystyle \int \operatorname {arctanh} x\,dx=x\operatorname {arctanh} x+{\frac {1}{2}}\log {(1-x^{2})}+C}$

${\displaystyle \int \operatorname {arccsch} \,x\,dx=x\operatorname {arccsch} x+\log {\left}+C}$

${\displaystyle \int \operatorname {arcsech} \,x\,dx=x\operatorname {arcsech} x-\arctan {\left({\frac {x}{x-1}}{\sqrt {\frac {1-x}{1+x}}}\right)}+C}$

${\displaystyle \int \operatorname {arccoth} \,x\,dx=x\operatorname {arccoth} x+{\frac {1}{2}}\log {(x^{2}-1)}+C}$