Prima metodă de schimbare de variabilă [ ] Fie I , J două intervale din
R
{\displaystyle \mathbb R \!}
I
⟶
φ
J
⟶
f
R
,
{\displaystyle I \overset {\varphi}{\longrightarrow} J \overset {f}{\longrightarrow} \mathbb R, }
cu proprietăţile:
a)
φ
{\displaystyle \varphi \!}
I .
b) f admite primitive; F este o primitivă a sa;
Atunci
(
f
∘
φ
)
⋅
φ
′
{\displaystyle (f \circ \varphi) \cdot \varphi' \!}
I şi
F
∘
φ
{\displaystyle F \circ \varphi}
(
f
∘
φ
)
⋅
φ
′
{\displaystyle (f \circ \varphi ) \cdot \varphi'}
∫
f
(
φ
(
x
)
)
⋅
φ
′
(
x
)
d
x
=
F
(
φ
(
x
)
)
+
C
.
{\displaystyle \int f(\varphi (x)) \cdot \varphi' (x) dx = F(\varphi (x)) + \mathcal C. \!}
A doua metodă de schimbare de variabilă [ ] Fie I , J două intervale din
R
{\displaystyle \mathbb R \!}
I
⟶
φ
J
⟶
f
R
,
{\displaystyle I \overset {\varphi}{\longrightarrow} J \overset {f}{\longrightarrow} \mathbb R, }
cu proprietăţile:
a)
φ
{\displaystyle \varphi \!}
I . b) funcţia
h
=
(
f
∘
φ
)
⋅
φ
′
{\displaystyle h= (f \circ \varphi) \cdot \varphi' \!}
H
⇒
{\displaystyle \Rightarrow \!}
f admite primitive şi funcţia
H
∘
φ
−
1
{\displaystyle H \circ \varphi^{-1} \!}
f .
∫
f
(
x
)
d
x
=
(
H
∘
φ
−
1
)
(
x
)
+
C
.
{\displaystyle \int f(x) dx = (H \circ \varphi^{-1}) (x) + \mathcal C.}
Formule [ ] Puterea lui t [ ]
∫
x
n
=
x
n
+
1
n
+
1
+
C
(
n
≠
−
1
)
{\displaystyle \int x^n = \frac {x^{n+1}}{n+1} + \mathcal C \; (n \neq -1) \!}
Demonstraţia 1 :
Avem
d
d
x
x
m
=
m
x
m
−
1
.
{\displaystyle \frac {d}{dx} x^m = mx^{m-1} .\!}
∫
m
x
m
−
1
=
x
m
+
C
{\displaystyle \int mx^{m-1} = x^m + \mathcal C \!}
m
=
n
+
1.
{\displaystyle m=n+1. \!}
Demonstraţia 2 : (Metoda lui Fermat )
Se ştie că
1
+
r
+
r
2
+
⋯
+
r
n
=
1
−
r
n
+
1
1
−
r
{\displaystyle 1+r+r^2 + \cdots + r^n = \frac {1-r^{n+1}}{1-r} \!}
şi
1
+
r
+
r
2
+
⋯
=
1
1
−
r
{\displaystyle 1+r + r^2 + \cdots = \frac {1}{1-r} \!}
r
<
1
{\displaystyle r<1 \!}
∫
0
b
x
n
d
x
{\displaystyle \int_0^b x^n dx \!}
r
↘
0
,
{\displaystyle r \searrow 0, \!}
∫
0
b
x
n
d
x
=
f
(
b
)
⋅
(
b
−
b
r
)
+
f
(
b
r
)
⋅
(
b
r
−
b
r
2
)
+
f
(
b
r
2
)
⋅
(
b
r
2
−
b
r
3
)
+
⋯
=
{\displaystyle \int_0^b x^n dx = f(b) \cdot (b-br) + f(br) \cdot (br-br^2) + f(br^2) \cdot (br^2-br^3) + \cdots = \!}
=
b
n
+
1
(
1
−
r
)
[
1
+
r
n
+
1
+
(
r
n
+
1
)
2
+
⋯
]
=
{\displaystyle = b^{n+1} (1-r) [ 1+ r^{n+1} + (r^{n+1})^2 + \cdots ] = \!}
=
b
n
+
1
(
1
−
r
)
[
1
1
−
r
n
+
1
]
=
b
n
+
1
1
+
r
+
r
2
+
⋯
+
r
n
=
b
n
+
1
n
+
1
.
{\displaystyle = b^{n+1} (1-r) [\frac {1}{1-r^{n+1}}] = \frac{b^{n+1}}{1+r + r^2 + \cdots + r^n} = \frac {b^{n+1}}{n+1}. \!}
QED .
Funcţia exponenţială şi cea logaritmică [ ]
∫
e
x
d
x
=
e
x
+
C
{\displaystyle \int e^xdx = e^x + \mathcal C \!}
Demonstraţie :
Se aplică relaţia
d
d
x
e
x
=
e
x
{\displaystyle \frac {d}{dx} e^x = e^x \!}
Aplicaţii [ ] Considerăm funcţia
f
:
[
0
,
1
]
×
[
0
,
1
]
→
R
2
{\displaystyle f: [0, 1] \times [0, 1] \rightarrow \mathbb R^2}
f
(
x
,
y
)
=
{
x
2
y
x
4
+
y
2
,
d
a
c
a
˘
(
x
,
y
)
≠
(
0
,
0
)
0
,
d
a
c
a
˘
(
x
,
y
)
=
(
0
,
0
)
{\displaystyle
f(x, y)=
\begin{cases}
\frac {x^2 y}{x^4 + y^2}, \; dac \breve{a} \; (x, y) \neq (0, 0)
\\
\\
0 , \; dac \breve{a} \; (x, y)= (0, 0)
\end{cases}
}
Această funcţie nu este integrabilă deoarece are un punct discontinuitate .
Considerăm funcţia
f
:
[
−
1
,
1
]
×
[
−
1
,
1
]
→
R
,
{\displaystyle f: [-1, 1] \times [-1, 1] \rightarrow \mathbb R,}
f
(
x
,
y
)
=
{
2
x
y
(
x
2
+
y
2
)
2
,
d
a
c
a
˘
(
x
,
y
)
≠
0
0
,
d
a
c
a
˘
(
x
,
y
)
=
(
0
,
0
)
{\displaystyle
f(x, y)=
\begin{cases}
\frac {2xy}{(x^2 + y^2)^2}, \; dac \breve{a} \; (x, y) \neq 0
\\
\\
0, \; dac \breve {a} \; (x, y) = (0, 0)
\end{cases}
}
Această funcţie nu este integrabilă deoarece nu este mărginită.
Calculul ariilor, lungimilor [ ]
Fie
f
:
[
a
,
b
]
→
R
,
{\displaystyle f: [a, b] \rightarrow \mathbb R, \!}
G graficul funcţiei continue.
A
(
G
)
−
∫
a
b
f
(
x
)
d
x
{\displaystyle A(G)- \int_a^b f(x)dx \!}
Lungimea graficului funţiei:
L
(
f
)
=
∫
a
b
1
+
(
f
′
(
x
)
)
2
{\displaystyle L(f) = \int_a^b \sqrt{1 + (f'(x))^2} \!}
Calculul volumelor [ ]
Aria suprafeţei de rotaţie
Fie
f
:
[
a
,
b
]
→
[
0
,
+
∞
)
{\displaystyle f:[a, b] \rightarrow [0, + \infty) \!}
A
(
f
)
=
2
π
∫
a
b
f
(
x
)
⋅
1
+
(
f
′
(
x
)
)
2
d
x
{\displaystyle A(f) = 2 \pi \int_a^b f(x) \cdot \sqrt {1 + (f'(x))^2} dx}
Volum corp de rotaţie:
V
(
f
)
=
∫
a
b
f
2
(
x
)
d
x
.
{\displaystyle V(f) = \int_a^b f^2(x) dx.}
Vezi şi [ ]
Resurse [ ] 
![{\displaystyle =b^{n+1}(1-r)[1+r^{n+1}+(r^{n+1})^{2}+\cdots ]=\!}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/2bbbfee39ad750ef073ba85432af38df594273d7)
![{\displaystyle =b^{n+1}(1-r)[{\frac {1}{1-r^{n+1}}}]={\frac {b^{n+1}}{1+r+r^{2}+\cdots +r^{n}}}={\frac {b^{n+1}}{n+1}}.\!}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/e7aca9eccc6469d57070c483595b7671b821a68e)
![{\displaystyle f:[0,1]\times [0,1]\rightarrow \mathbb {R} ^{2}}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/6a1b59321c20390fd05b316a4b8f9556d46e5aad)
![{\displaystyle f:[-1,1]\times [-1,1]\rightarrow \mathbb {R} ,}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/8fadc8bd3f2cf8a583dcf3c8f35d0ac4ff71092b)
![{\displaystyle f:[a,b]\rightarrow \mathbb {R} ,\!}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/e594637eeb1c7238a2dabb4f9295d18f5bf40946)
![{\displaystyle f:[a,b]\rightarrow [0,+\infty )\!}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/38104e31fb9dde45db6a20cf5327f56ad5c35f77)
