Metoda supstitucije je metoda rešavanja integrala u kojoj se deo integrala zamenjuje jednostavnijim simbolom (obično se koristi latinično slovo u), u cilju da bi se dobio integral koji je lakše rešiti.[ 1]
∫
ln
x
x
d
x
=
?
{\displaystyle \int {\frac {\ln {x}{x}dx=?}
u
=
l
n
(
x
)
{\displaystyle u=ln(x)}
d
u
=
1
x
d
x
{\displaystyle du={\frac {1}{x}dx}
x
d
u
=
d
x
{\displaystyle xdu=dx}
∫
l
n
(
x
)
x
d
x
=
∫
u
x
x
d
u
=
u
2
2
+
C
{\displaystyle \int {\frac {ln(x)}{x}dx=\int {\frac {u}{x}xdu={\frac {u^{2}{2}+C}
∫
l
n
(
x
)
x
d
x
=
(
ln
(
x
)
)
2
2
+
C
{\displaystyle \int {\frac {ln(x)}{x}dx={\frac {(\ln(x))^{2}{2}+C}
U nekim slučajevima moraće se rešiti za
x
{\displaystyle x}
∫
2
x
5
x
−
20
d
x
=
?
{\displaystyle \int 2x{\sqrt {5x-20}dx=?}
ω
=
5
x
−
20
{\displaystyle \omega =5x-20}
d
ω
=
5
d
x
{\displaystyle d\omega =5dx}
d
ω
5
=
d
x
{\displaystyle {\frac {d\omega }{5}=dx}
x
=
ω
+
20
5
{\displaystyle x={\frac {\omega +20}{5}
∫
2
x
5
x
−
20
d
x
=
∫
2
(
ω
+
20
5
)
ω
d
ω
5
=
2
25
∫
(
ω
+
20
)
ω
d
ω
=
2
25
∫
(
ω
ω
+
20
ω
)
d
ω
=
2
25
(
∫
ω
ω
d
ω
+
∫
20
ω
d
ω
)
{\displaystyle \int 2x{\sqrt {5x-20}dx=\int 2\left({\frac {\omega +20}{5}\right){\sqrt {\omega }{\frac {d\omega }{5}={\frac {2}{25}\int (\omega +20){\sqrt {\omega }d\omega ={\frac {2}{25}\int (\omega {\sqrt {\omega }+20{\sqrt {\omega })d\omega ={\frac {2}{25}\left(\int \omega {\sqrt {\omega }d\omega +\int 20{\sqrt {\omega }d\omega \right)}
2
25
(
∫
ω
ω
d
ω
+
∫
20
ω
d
ω
)
=
2
25
(
ω
5
2
5
2
+
20
ω
3
2
3
2
)
+
C
{\displaystyle {\frac {2}{25}\left(\int \omega {\sqrt {\omega }d\omega +\int 20{\sqrt {\omega }d\omega \right)={\frac {2}{25}\left({\frac {\omega ^{\frac {5}{2}{\frac {5}{2}+20{\frac {\omega ^{\frac {3}{2}{\frac {3}{2}\right)+C}
∫
2
x
5
x
−
20
d
x
=
2
25
(
(
5
x
−
20
)
5
2
5
2
+
20
(
5
x
−
20
)
3
2
3
2
)
+
C
{\displaystyle \int 2x{\sqrt {5x-20}dx={\frac {2}{25}\left({\frac {(5x-20)^{\frac {5}{2}{\frac {5}{2}+20{\frac {(5x-20)^{\frac {3}{2}{\frac {3}{2}\right)+C}
Metoda supstitucije se može koristiti za definisanje novih antiderivata.
∫
tan
x
d
x
=
?
{\displaystyle \int \tan {x}dx=?}
∫
tan
x
d
x
=
∫
sin
x
cos
x
d
x
{\displaystyle \int \tan {x}dx=\int {\frac {\sin {x}{\cos {x}dx}
u
=
cos
x
{\displaystyle u=\cos {x}
d
u
=
−
sin
x
d
x
{\displaystyle du=-\sin {x}dx}
d
u
−
s
i
n
x
=
d
x
{\displaystyle {\frac {du}{-sin{x}=dx}
∫
sin
x
cos
x
d
x
=
∫
s
i
n
x
u
d
u
−
sin
x
=
−
∫
1
u
d
u
=
−
ln
u
+
C
{\displaystyle \int {\frac {\sin {x}{\cos {x}dx=\int {\frac {sin{x}{u}{\frac {du}{-\sin {x}=-\int {\frac {1}{u}du=-\ln {u}+C}
∫
tan
x
d
x
=
ln
c
o
s
x
+
C
{\displaystyle \int \tan {x}dx=\ln {cos{x}+C}
Reference
Literatura
Briggs, William; Cochran, Lyle (2011), Calculus /Early Transcendentals (Single Variable изд.), Addison-Wesley, ISBN 978-0-321-66414-3 Ferzola, Anthony P. (1994), „Euler and differentials” , The College Mathematics Journal , 25 (2): 102–111, JSTOR 2687130 , doi :10.2307/2687130 , Архивирано из оригинала 07. 11. 2012. г., Приступљено 13. 11. 2022 Fremlin, D.H. (2010), Measure Theory, Volume 2 , Torres Fremlin, ISBN 978-0-9538129-7-4 Hewitt, Edwin; Stromberg, Karl (1965), Real and Abstract Analysis , Springer-Verlag, ISBN 978-0-387-04559-7 Katz, V. (1982), „Change of variables in multiple integrals: Euler to Cartan”, Mathematics Magazine , 55 (1): 3–11, JSTOR 2689856 , doi :10.2307/2689856 Rudin, Walter (1987), Real and Complex Analysis , McGraw-Hill, ISBN 978-0-07-054234-1 Swokowski, Earl W. (1983), Calculus with analytic geometry (alternate изд.), Prindle, Weber & Schmidt, ISBN 0-87150-341-7 Spivak, Michael (1965), Calculus on Manifolds , Westview Press, ISBN 978-0-8053-9021-6
Numerička integracija
Riemann integral
Lebesgue integral
Burkill integral
Bochner integral
Daniell integral
Darboux integral
Henstock–Kurzweil integral
Harov integral Hellinger integral
Khinchin integral
Kolmogorov integral
Lebesgue–Stieltjes integral
Pettis integral
Pfeffer integral
Riemann–Stieltjes integral
Regulated integral Metode
Integration by parts
Integration by substitution
Inverse function integration
Order of integration (calculus)
trigonometric substitution
Integration by partial fractions
Integration by reduction formulae
Integration using parametric derivatives
Integration using Euler's formula
Differentiation under the integral sign
Metode konturne integracije Nesvojstveni integral Stohastički integrali
Itô integral
Stratonovich integral
Skorokhod integral
