📐 Integral Hisob — Nazariya

Kirish

Integral hisob — differensial hisobning teskari amali. Robotika, dronlar va raketalar uchun masofa, ish, energiya va boshqa kumulyativ kattaliklar hisoblashda zarur.

1. Aniqmas Integral (Indefinite Integral)

Ta'rif

$F(x)$ funksiya $f(x)$ ning antiderivativi (primitive funksiya), agar:

$F'(x) = f(x)$

Aniqmas integral:

$\int f(x) \, dx = F(x) + C$

bu yerda $C$ — integrallash konstantasi.

Asosiy integrallar

Funksiya $f(x)$	Integral $\int f(x)dx$
$x^n$ ($n \neq -1$)	$\frac{x^{n+1}}{n+1} + C$
$\frac{1}{x}$	$\ln
$e^x$	$e^x + C$
$a^x$	$\frac{a^x}{\ln a} + C$
$\sin x$	$-\cos x + C$
$\cos x$	$\sin x + C$
$\sec^2 x$	$\tan x + C$
$\csc^2 x$	$-\cot x + C$
$\frac{1}{\sqrt{1-x^2}}$	$\arcsin x + C$
$\frac{1}{1+x^2}$	$\arctan x + C$

2. Integrallash qoidalari

Asosiy qoidalar

1. Konstanta: $\int k \cdot f(x) \, dx = k \int f(x) \, dx$

2. Yig'indi: $\int [f(x) + g(x)] \, dx = \int f(x) \, dx + \int g(x) \, dx$

3. Chiziqlilik: $\int [af(x) + bg(x)] \, dx = a\int f(x) \, dx + b\int g(x) \, dx$

3. Integrallash usullari

3.1 Almashtiruv usuli (Substitution)

Agar $u = g(x)$ bo'lsa:

$\int f(g(x)) \cdot g'(x) \, dx = \int f(u) \, du$

Misol

$\int 2x \cdot e^{x^2} dx$

$u = x^2$, $du = 2x \, dx$

$\int e^u \, du = e^u + C = e^{x^2} + C$

3.2 Qismlab integrallash (Integration by Parts)

$\int u \, dv = uv - \int v \, du$

LIATE qoidasi — $u$ ni tanlash tartibi:

• Logarifm: $\ln x$
• Inverse trig: $\arctan x$, $\arcsin x$
• Algebraik: $x^n$, polinomlar
• Trigonometrik: $\sin x$, $\cos x$
• Eksponensial: $e^x$

Misol

$\int x \cdot e^x dx$

$u = x$, $dv = e^x dx$ $du = dx$, $v = e^x$

$\int x e^x dx = xe^x - \int e^x dx = xe^x - e^x + C = e^x(x-1) + C$

3.3 Trigonometrik almashtiruv

Ifoda	Almashtiruv
$\sqrt{a^2 - x^2}$	$x = a\sin\theta$
$\sqrt{a^2 + x^2}$	$x = a\tan\theta$
$\sqrt{x^2 - a^2}$	$x = a\sec\theta$

3.4 Qisman kasr usuli (Partial Fractions)

Ratsional funksiyalar uchun:

$\frac{P(x)}{Q(x)} = \frac{A}{x-a} + \frac{B}{x-b} + ...$

4. Aniq Integral (Definite Integral)

Ta'rif

$\int_a^b f(x) \, dx = F(b) - F(a)$

bu yerda $F(x)$ — $f(x)$ ning antiderivativi.

Geometrik ma'no

Aniq integral — egri chiziq ostidagi yuza:

    y
    |     _____
    |    /     \
    |   /       \   ← f(x)
    |  /  yuza   \
    | /___________\
    +-----|-----|----→ x
          a     b

$\text{Yuza} = \int_a^b f(x) \, dx$

Xossalar

1. Chegaralarni almashtirish: $\int_a^b f(x) \, dx = -\int_b^a f(x) \, dx$

2. Qo'shish: $\int_a^b f(x) \, dx + \int_b^c f(x) \, dx = \int_a^c f(x) \, dx$

3. Chiziqlilik: $\int_a^b [cf(x) + dg(x)] \, dx = c\int_a^b f(x) \, dx + d\int_a^b g(x) \, dx$

5. Hisob asosiy teoremasi

Birinchi qism

Agar $f$ uzluksiz bo'lsa:

$\frac{d}{dx}\int_a^x f(t) \, dt = f(x)$

Ikkinchi qism (Newton-Leibniz)

$\int_a^b f(x) \, dx = F(b) - F(a)$

6. Yuzalar va hajmlar

Egri chiziqlar orasidagi yuza

$A = \int_a^b |f(x) - g(x)| \, dx$

Aylanish jismi hajmi

X o'qi atrofida: $V = \pi \int_a^b [f(x)]^2 \, dx$

Disk usuli: $V = \pi \int_a^b [R(x)^2 - r(x)^2] \, dx$

Yoy uzunligi

$L = \int_a^b \sqrt{1 + [f'(x)]^2} \, dx$

7. Noto'g'ri integrallar

Cheksiz chegaralar

$\int_a^\infty f(x) \, dx = \lim_{b \to \infty} \int_a^b f(x) \, dx$

Uzilish nuqtasi

Agar $f(x)$ $x = c$ da uzilsa:

$\int_a^b f(x) \, dx = \lim_{\epsilon \to 0^+} \left[\int_a^{c-\epsilon} + \int_{c+\epsilon}^b\right]$

8. Ko'p o'lchovli integrallar

Ikki karrali integral

$\iint_R f(x, y) \, dA = \int_a^b \int_{c}^{d} f(x, y) \, dy \, dx$

Uch karrali integral

$\iiint_V f(x, y, z) \, dV$

Polar koordinatalar

$\iint f(r, \theta) \, r \, dr \, d\theta$

9. Fizik qo'llanmalar

Ish (Work)

O'zgaruvchan kuch bajargan ish:

$W = \int_a^b F(x) \, dx$

Impuls

$J = \int_{t_1}^{t_2} F(t) \, dt = \Delta p$

Massa markazi

$\bar{x} = \frac{\int x \cdot \rho(x) \, dx}{\int \rho(x) \, dx}$

Inersiya momenti

$I = \int r^2 \, dm$

10. Robotika va raketada qo'llanish

Tezlikdan masofa

$s = \int_{t_0}^{t_1} v(t) \, dt$

Tezlanishdan tezlik

$v = v_0 + \int_{t_0}^{t_1} a(t) \, dt$

Raketa massasi

$m(t) = m_0 - \int_0^t \dot{m}(\tau) \, d\tau$

PID integral termi

$u_I = K_i \int_0^t e(\tau) \, d\tau$

Energiya sarfi

$E = \int_0^T P(t) \, dt$

11. Raqamli integrallash

Trapetsiya qoidasi

$\int_a^b f(x) \, dx \approx \frac{h}{2}[f(a) + 2f(x_1) + ... + 2f(x_{n-1}) + f(b)]$

Simpson qoidasi

$\int_a^b f(x) \, dx \approx \frac{h}{3}[f(a) + 4f(x_1) + 2f(x_2) + 4f(x_3) + ... + f(b)]$

Xulosa

Integral hisob asoslari:

• ✅ Aniqmas va aniq integrallar
• ✅ Integrallash usullari (almashtiruv, qismlab)
• ✅ Yuzalar va hajmlar
• ✅ Noto'g'ri integrallar
• ✅ Fizik qo'llanmalar
• ✅ Raqamli integrallash