İntegral

Qısımo ke binê fonksiyon dero u herfa S ra nusiyao , İntegralo

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

Formulê cı

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

$S=\underset{\mathrm{\Delta }x\to 0}{\lim }\sum _{i=0}^{n-1}f(x_i)\mathrm{\Delta }{x}_i=\underset{a}{\overset{b}{\int }}f(x)dx=F(b)-F(a)$

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:

$x^6-9x^3+8=0.$

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

$u^2-9u+8=0$

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

$u=1\text{ve}u=8^{\prime }dir.$

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

$x^3=1\text{ve}{x}^3=8\Rightarrow x=(1{)}^{1/3}=1\text{ve}x=(8{)}^{1/3}=2.$

İntegralê fonksiyonê basiti

Fonksiyonê rasyonali

$\int dx=x+C$

$\int x^n\mathrm{d}x=\frac{x^{n+1}}{n+1}+C\text{ eğer }n\ne -1$

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

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

Fonksiyonê irrasyonali

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

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

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

Fonksiyonê logaritmiki

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

$\int {\log }_{b}xdx=x{\log }_{b}x-x{\log }_{b}e+C$

Fonskyionê ke serê cı esto

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

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

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

Fonksiyonê trigonometriki

$\int \sin xdx=-\cos x+C$

$\int \cos xdx=\sin x+C$

$\int \tan xdx=-\ln |\cos x|+C$

$\int \cot xdx=\ln |\sin x|+C$

$\int \sec xdx=\ln |\sec x+\tan x|+C$

$\int \csc xdx=\ln |\csc x-\cot x|+C$

$\int {\sec }^{2}xdx=\tan x+C$

$\int {\csc }^{2}xdx=-\cot x+C$

$\int \sec x\tan xdx=\sec x+C$

$\int \csc x\cot xdx=-\csc x+C$

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

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

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

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

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

$\int \arctan xdx=x\arctan x-\frac{1}{2}\ln |1+x^2|+C$

Fonksiyonê hiperboliki

$\int \sinh xdx=-coshx+C$

$\int \cosh xdx=\sinh x+C$

$\int \tanh xdx=\ln |\cosh x|+C$

$\int \text{csch}xdx=\ln |\tanh \frac{x}{2}|+C$

$\int \text{sech}xdx=\arctan (\sinh x)+C$

$\int \coth xdx=\ln |\sinh x|+C$

$\int {\text{sech}}^{2}xdx=\tanh x+C$

Fonksiyonê hiperbolikio ters

$\int \mathrm{arcsinh}xdx=x\mathrm{arcsinh}x-\sqrt{x^2+1}+C$

$\int \mathrm{arccosh}xdx=x\mathrm{arccosh}x-\sqrt{x^2-1}+C$

$\int \mathrm{arctanh}xdx=x\mathrm{arctanh}x+\frac{1}{2}\log (1-x^2)+C$

$\int \mathrm{arccsch}xdx=x\mathrm{arccsch}x+\log \left[x(\sqrt{1+\frac{1}{x^2}}+1)\right]+C$

$\int \mathrm{arcsech}xdx=x\mathrm{arcsech}x-\arctan \left(\frac{x}{x-1}\sqrt{\frac{1-x}{1+x}}\right)+C$

$\int \mathrm{arccoth}xdx=x\mathrm{arccoth}x+\frac{1}{2}\log (x^2-1)+C$