F(x) . f(x)
() = ∫ f(t)dt .
'(x) = f(x), .. ∫ f(t)dt = F(x) + C
1
=,
1 = F(a)
C- ∫ f(x)dx = F(x) - F(a)
x=b, ∫ f(t)dt = F(b) - F(a)
-
.
- . -. . =t - =j(t), j(t) . . - [a;b] j(a)=a, j(β)=b. . -:
∫ f(x)dx = ∫ f[φ(t)] * φ'(t) dt, φ'(t) dt = dx
-.
F(x) . - f(x). :
∫ f(x)dx = F(x) + C
∫ f[φ(t)] * φ'(t) dt = F[φ(t)] + C
∫ f(x)dx =F(x) │= F(b)-F(a), ∫ f[φ(t)]*φ'(t)dt =F[φ(t)] │= F[j(β)] - F[j(a)]= F(b) - F(a)
. 2- . - , , . . - , . - . -.
.
-. (u*v)' = u'v + v'u, v,u - -.
a b:
∫ (u*v)dx = ∫ u'v dx + ∫ uv' dx
.. ∫ (u*v)'dx = uv + C, ∫ (u*v)'dx = uv │,
∫ u dv = uv│ - ∫ v du, du = u' dx
.
[a;b] - y = f(x) .
S. ., . ,
=, =b.
Q = ∫ f(x)dx
f(x)≤0, -Q = ∫ f(x)dx, Q = - ∫ f(x)dx
- - . [a;b], - . - . . ,
f(x) >0, , f(x)<0.
, ,
Q = ∫ │f(x)│dx
, . y = f
1(x) y=f
2(x), f
1(x) £ f
2(x) [a;b],
Q = ∫ [ f2(x) - f1(x) ]dx
.
- . - . - .
. ∫ f(x)dx =2 ∫ f(x)dx, Q=2 ∫ xdx
, . ,
, .. y = ψ(t), α ≤ t ≤ β
a = φ(α), b = φ(β) x = φ(t)
- . . -. y=f(x).
. , . t.
. Q = ∫ y dx Q = ∫ ψ(t) * ψ'(t) dt