Seba fonksiyonê deriativ vıraştış asêniye dano.
Qısımo ke binê fonksiyon dero u herfa S ra nusiyao , İntegral o
F ( x ) = ∫ f ( x ) + c , {\displaystyle F(x)=\int f(x)+c,} o. Raya Limiti ra formulê ho u pêrokerdışê Riemanni ; S = lim Δ x → 0 ∑ i = 0 n − 1 f ( x i ) Δ x i = ∫ a b f ( x ) d x = F ( b ) − F ( a ) {\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:
x 6 − 9 x 3 + 8 = 0. {\displaystyle x^{6}-9x^{3}+8=0.\,} Ena denkleme de x 3 = u ebe vurnayışê varyanti denlkem bin de ma nuseni ;
u 2 − 9 u + 8 = 0 {\displaystyle u^{2}-9u+8=0\,} Ena raya ra denklem derecey 2. ra biya. Rıstımê enay denkleme ;
u = 1 ve u = 8 ′ d i r . {\displaystyle u=1\quad {\mbox{ve}}\quad u=8'dir.} Ebe ena vurnayışo newa ra netıcey varyantê esasi de cay ho ra bınusi ;
x 3 = 1 ve x 3 = 8 ⇒ x = ( 1 ) 1 / 3 = 1 ve x = ( 8 ) 1 / 3 = 2. {\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 ∫ d x = x + C {\displaystyle \int dx=x+C} ∫ x n d x = x n + 1 n + 1 + C eğer n ≠ − 1 {\displaystyle \int x^{n}\,{\rm {d}}x={\frac {x^{n+1}}{n+1}}+C\qquad {\mbox{ eğer }}n\neq -1} ∫ d x x = ln | x | + C {\displaystyle \int {dx \over x}=\ln {\left|x\right|}+C} ∫ d x a 2 + x 2 = 1 a arctan x a + C {\displaystyle \int {dx \over {a^{2}+x^{2}}}={1 \over a}\arctan {x \over a}+C} Fonksiyonê irrasyonali ∫ d x a 2 − x 2 = arcsin x a + C {\displaystyle \int {dx \over {\sqrt {a^{2}-x^{2}}}}=\arcsin {x \over a}+C} ∫ − d x a 2 − x 2 = arccos x a + C {\displaystyle \int {-dx \over {\sqrt {a^{2}-x^{2}}}}=\arccos {x \over a}+C} ∫ d x x x 2 − a 2 = 1 a sec | x | a + C {\displaystyle \int {dx \over x{\sqrt {x^{2}-a^{2}}}}={1 \over a}\sec {|x| \over a}+C} ∫ ln ( x ) d x = x ln ( x ) − x + C , {\displaystyle \int \ln(x)\,dx=x\ln(x)-x+C,} ∫ log b x d x = x log b x − x log b e + C {\displaystyle \int \log _{b}{x}\,dx=x\log _{b}{x}-x\log _{b}{e}+C} Fonskyionê ke serê cı esto ∫ 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} ∫ a l n ( x ) d x = ∫ x l n ( a ) d x = x a l n ( x ) ln a + 1 + C = x x l n ( a ) ln a + 1 + 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 ∫ 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 | cos x | + C {\displaystyle \int \tan {x}\,dx=-\ln {\left|\cos {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} ∫ csc x d x = ln | csc x − cot x | + C {\displaystyle \int \csc {x}\,dx=\ln {\left|\csc {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 2 x d x = 1 2 ( x − sin x cos x ) + C {\displaystyle \int \sin ^{2}x\,dx={\frac {1}{2}}(x-\sin x\cos x)+C} ∫ cos 2 x d x = 1 2 ( x + sin x cos x ) + C {\displaystyle \int \cos ^{2}x\,dx={\frac {1}{2}}(x+\sin x\cos x)+C} ∫ sec 3 x d x = 1 2 sec x tan x + 1 2 ln | sec x + tan x | + C {\displaystyle \int \sec ^{3}x\,dx={\frac {1}{2}}\sec x\tan x+{\frac {1}{2}}\ln |\sec x+\tan x|+C} ∫ 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} ∫ arctan x d x = x arctan x − 1 2 ln | 1 + x 2 | + C {\displaystyle \int \arctan {x}\,dx=x\,\arctan {x}-{\frac {1}{2}}\ln {\left|1+x^{2}\right|}+C} Fonksiyonê hiperboliki ∫ sinh x d x = − c o s h x + C {\displaystyle \int \sinh x\,dx=-\,coshx+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} Fonksiyonê hiperbolikio ters ∫ arcsinh x d x = x arcsinh x − x 2 + 1 + C {\displaystyle \int \operatorname {arcsinh} x\,dx=x\operatorname {arcsinh} x-{\sqrt {x^{2}+1}}+C} ∫ arccosh x d x = x arccosh x − x 2 − 1 + C {\displaystyle \int \operatorname {arccosh} x\,dx=x\operatorname {arccosh} x-{\sqrt {x^{2}-1}}+C} ∫ arctanh x d x = x arctanh x + 1 2 log ( 1 − x 2 ) + C {\displaystyle \int \operatorname {arctanh} x\,dx=x\operatorname {arctanh} x+{\frac {1}{2}}\log {(1-x^{2})}+C} ∫ arccsch x d x = x arccsch x + log [ x ( 1 + 1 x 2 + 1 ) ] + C {\displaystyle \int \operatorname {arccsch} \,x\,dx=x\operatorname {arccsch} x+\log {\left[x\left({\sqrt {1+{\frac {1}{x^{2}}}}}+1\right)\right]}+C} ∫ arcsech x d x = x arcsech x − arctan ( x x − 1 1 − x 1 + x ) + 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} ∫ arccoth x d x = x arccoth x + 1 2 log ( x 2 − 1 ) + C {\displaystyle \int \operatorname {arccoth} \,x\,dx=x\operatorname {arccoth} x+{\frac {1}{2}}\log {(x^{2}-1)}+C}
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