✏️ Integral Hisob — Masalalar

1-bo'lim: Asosiy integrallar (5 ta masala)

Masala 1.1 ⭐⭐

Integralni hisoblang: $\int (3x^2 - 4x + 5) \, dx$

Yechim

Har bir hadni alohida integrallaymiz:

$\int 3x^2 dx - \int 4x dx + \int 5 dx$

$= 3 \cdot \frac{x^3}{3} - 4 \cdot \frac{x^2}{2} + 5x + C$

$= x^3 - 2x^2 + 5x + C$

Javob: $x^3 - 2x^2 + 5x + C$

Masala 1.2 ⭐⭐

$\int \frac{1}{\sqrt{x}} \, dx$

Yechim

$\int x^{-1/2} dx = \frac{x^{1/2}}{1/2} + C = 2\sqrt{x} + C$

Javob: $2\sqrt{x} + C$

Masala 1.3 ⭐⭐

$\int (e^x + \sin x) \, dx$

Yechim

$\int e^x dx + \int \sin x dx = e^x - \cos x + C$

Javob: $e^x - \cos x + C$

Masala 1.4 ⭐⭐

$\int \frac{3}{x} \, dx$

Yechim

$3 \int \frac{1}{x} dx = 3\ln|x| + C$

Javob: $3\ln|x| + C$

Masala 1.5 ⭐⭐⭐

$\int \sec^2 x + \frac{1}{1+x^2} \, dx$

Yechim

$\int \sec^2 x dx + \int \frac{1}{1+x^2} dx = \tan x + \arctan x + C$

Javob: $\tan x + \arctan x + C$

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2-bo'lim: Almashtiruv usuli (6 ta masala)

Masala 2.1 ⭐⭐

$\int 2x \cdot (x^2 + 1)^5 \, dx$

Yechim

$u = x^2 + 1$, $du = 2x \, dx$

$\int u^5 du = \frac{u^6}{6} + C = \frac{(x^2+1)^6}{6} + C$

Javob: $\frac{(x^2+1)^6}{6} + C$

Masala 2.2 ⭐⭐

$\int \cos(3x) \, dx$

Yechim

$u = 3x$, $du = 3dx$, $dx = \frac{du}{3}$

$\int \cos u \cdot \frac{du}{3} = \frac{1}{3}\sin u + C = \frac{\sin(3x)}{3} + C$

Javob: $\frac{\sin(3x)}{3} + C$

Masala 2.3 ⭐⭐⭐

$\int \frac{e^{\sqrt{x}}}{\sqrt{x}} \, dx$

Yechim

$u = \sqrt{x}$, $du = \frac{1}{2\sqrt{x}}dx$, $\frac{dx}{\sqrt{x}} = 2du$

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

Javob: $2e^{\sqrt{x}} + C$

Masala 2.4 ⭐⭐⭐

$\int \frac{x}{x^2 + 4} \, dx$

Yechim

$u = x^2 + 4$, $du = 2x \, dx$, $x \, dx = \frac{du}{2}$

$\int \frac{1}{u} \cdot \frac{du}{2} = \frac{1}{2}\ln|u| + C = \frac{\ln(x^2+4)}{2} + C$

Javob: $\frac{\ln(x^2+4)}{2} + C$

Masala 2.5 ⭐⭐⭐

$\int \tan x \, dx$

Yechim

$\int \frac{\sin x}{\cos x} dx$

$u = \cos x$, $du = -\sin x \, dx$

$-\int \frac{du}{u} = -\ln|u| + C = -\ln|\cos x| + C = \ln|\sec x| + C$

Javob: $\ln|\sec x| + C$

Masala 2.6 ⭐⭐⭐

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

Yechim

$u = -x^2$, $du = -2x \, dx$, $x \, dx = -\frac{du}{2}$

$-\frac{1}{2}\int e^u du = -\frac{e^u}{2} + C = -\frac{e^{-x^2}}{2} + C$

Javob: $-\frac{e^{-x^2}}{2} + C$

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3-bo'lim: Qismlab integrallash (5 ta masala)

Masala 3.1 ⭐⭐⭐

$\int x \cdot \cos x \, dx$

Yechim

$u = x$, $dv = \cos x \, dx$

$du = dx$, $v = \sin x$

$\int x \cos x dx = x\sin x - \int \sin x dx = x\sin x + \cos x + C$

Javob: $x\sin x + \cos x + C$

Masala 3.2 ⭐⭐⭐

$\int x^2 \cdot e^x \, dx$

Yechim

Ikki marta qismlab:

1. $u = x^2$, $dv = e^x dx$ → $du = 2x dx$, $v = e^x$

$x^2 e^x - 2\int x e^x dx$

2. $\int x e^x dx$: $u = x$, $dv = e^x dx$

$x e^x - e^x$

Natija: $x^2 e^x - 2(xe^x - e^x) + C = e^x(x^2 - 2x + 2) + C$

Javob: $e^x(x^2 - 2x + 2) + C$

Masala 3.3 ⭐⭐⭐

$\int \ln x \, dx$

Yechim

$u = \ln x$, $dv = dx$

$du = \frac{dx}{x}$, $v = x$

$x\ln x - \int x \cdot \frac{dx}{x} = x\ln x - x + C = x(\ln x - 1) + C$

Javob: $x(\ln x - 1) + C$

Masala 3.4 ⭐⭐⭐⭐

$\int e^x \sin x \, dx$

Yechim

$I = \int e^x \sin x dx$

1. $u = \sin x$, $dv = e^x dx$

$I = e^x \sin x - \int e^x \cos x dx$

2. $\int e^x \cos x dx$: $u = \cos x$, $dv = e^x dx$

$= e^x \cos x + \int e^x \sin x dx = e^x \cos x + I$

Qo'yamiz: $I = e^x \sin x - e^x \cos x - I$ $2I = e^x(\sin x - \cos x)$ $I = \frac{e^x(\sin x - \cos x)}{2} + C$

Javob: $\frac{e^x(\sin x - \cos x)}{2} + C$

Masala 3.5 ⭐⭐⭐⭐

$\int x \cdot \arctan x \, dx$

Yechim

$u = \arctan x$, $dv = x \, dx$

$du = \frac{dx}{1+x^2}$, $v = \frac{x^2}{2}$

$\frac{x^2}{2}\arctan x - \int \frac{x^2}{2(1+x^2)} dx$

$= \frac{x^2}{2}\arctan x - \frac{1}{2}\int \frac{x^2}{1+x^2} dx$

$= \frac{x^2}{2}\arctan x - \frac{1}{2}\int \left(1 - \frac{1}{1+x^2}\right) dx$

$= \frac{x^2}{2}\arctan x - \frac{x}{2} + \frac{\arctan x}{2} + C$

$= \frac{(x^2+1)\arctan x - x}{2} + C$

Javob: $\frac{(x^2+1)\arctan x - x}{2} + C$

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4-bo'lim: Aniq integrallar (5 ta masala)

Masala 4.1 ⭐⭐

$\int_0^2 (3x^2 + 1) \, dx$

Yechim

$[x^3 + x]_0^2 = (8 + 2) - (0 + 0) = 10$

Javob: $10$

Masala 4.2 ⭐⭐

$\int_0^{\pi/2} \cos x \, dx$

Yechim

$[\sin x]_0^{\pi/2} = \sin(\pi/2) - \sin(0) = 1 - 0 = 1$

Javob: $1$

Masala 4.3 ⭐⭐⭐

$\int_1^e \frac{1}{x} \, dx$

Yechim

$[\ln x]_1^e = \ln e - \ln 1 = 1 - 0 = 1$

Javob: $1$

Masala 4.4 ⭐⭐⭐

$\int_0^1 x \cdot e^{x^2} \, dx$

Yechim

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

Chegaralar: $x=0 \to u=0$, $x=1 \to u=1$

$\frac{1}{2}\int_0^1 e^u du = \frac{1}{2}[e^u]_0^1 = \frac{e-1}{2}$

Javob: $\frac{e-1}{2} \approx 0.859$

Masala 4.5 ⭐⭐⭐⭐

$\int_0^{\pi} x \cdot \sin x \, dx$

Yechim

Qismlab: $u = x$, $dv = \sin x dx$

$[{-x\cos x}]_0^{\pi} + \int_0^{\pi} \cos x dx$

$= (-\pi \cdot (-1) - 0) + [\sin x]_0^{\pi}$

$= \pi + (0 - 0) = \pi$

Javob: $\pi$

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5-bo'lim: Yuzalar (4 ta masala)

Masala 5.1 ⭐⭐⭐

$y = x^2$ va $y = 4$ orasidagi yuza.

Yechim

Kesishish nuqtalari: $x^2 = 4$ → $x = \pm 2$

$A = \int_{-2}^{2} (4 - x^2) dx = [4x - \frac{x^3}{3}]_{-2}^{2}$

$= (8 - \frac{8}{3}) - (-8 + \frac{8}{3}) = 16 - \frac{16}{3} = \frac{32}{3}$

Javob: $\frac{32}{3} \approx 10.67$ birlik kvadrat

Masala 5.2 ⭐⭐⭐

$y = \sin x$ va x-o'qi orasidagi yuza, $0$ dan $\pi$ gacha.

Yechim

$A = \int_0^{\pi} |\sin x| dx = \int_0^{\pi} \sin x dx$

$= [-\cos x]_0^{\pi} = -(-1) - (-1) = 2$

Javob: $2$ birlik kvadrat

Masala 5.3 ⭐⭐⭐⭐

$y = x$ va $y = x^2$ orasidagi yuza.

Yechim

Kesishish: $x = x^2$ → $x(x-1) = 0$ → $x = 0, 1$

$[0, 1]$ oralig'ida $x \geq x^2$

$A = \int_0^1 (x - x^2) dx = [\frac{x^2}{2} - \frac{x^3}{3}]_0^1 = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}$

Javob: $\frac{1}{6} \approx 0.167$ birlik kvadrat

Masala 5.4 ⭐⭐⭐⭐

$y = e^x$, $y = 1$, $x = 0$, $x = 1$ orasidagi yuza.

Yechim

$[0, 1]$ oralig'ida $e^x \geq 1$

$A = \int_0^1 (e^x - 1) dx = [e^x - x]_0^1 = (e - 1) - (1 - 0) = e - 2$

Javob: $e - 2 \approx 0.718$ birlik kvadrat

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6-bo'lim: Hajmlar (4 ta masala)

Masala 6.1 ⭐⭐⭐

$y = \sqrt{x}$, $0 \leq x \leq 4$ ni x-o'qi atrofida aylantiring. Hajmni toping.

Yechim

$V = \pi \int_0^4 (\sqrt{x})^2 dx = \pi \int_0^4 x dx = \pi [\frac{x^2}{2}]_0^4 = \pi \cdot 8 = 8\pi$

Javob: $8\pi \approx 25.13$ birlik kub

Masala 6.2 ⭐⭐⭐

$y = x^2$, $0 \leq x \leq 2$ ni x-o'qi atrofida aylantiring.

Yechim

$V = \pi \int_0^2 (x^2)^2 dx = \pi \int_0^2 x^4 dx = \pi [\frac{x^5}{5}]_0^2 = \frac{32\pi}{5}$

Javob: $\frac{32\pi}{5} \approx 20.11$ birlik kub

Masala 6.3 ⭐⭐⭐⭐

Shar hajmi ($R$ radiusli).

Yechim

Shar: $y = \sqrt{R^2 - x^2}$ ni x-o'qi atrofida aylantirish.

$V = \pi \int_{-R}^{R} (R^2 - x^2) dx = \pi [R^2x - \frac{x^3}{3}]_{-R}^{R}$

$= \pi [(R^3 - \frac{R^3}{3}) - (-R^3 + \frac{R^3}{3})]$

$= \pi \cdot 2(R^3 - \frac{R^3}{3}) = \frac{4\pi R^3}{3}$

Javob: $V = \frac{4\pi R^3}{3}$

Masala 6.4 ⭐⭐⭐⭐

Konus hajmi (balandlik $h$, radius $R$).

Yechim

To'g'ri chiziq: $y = \frac{R}{h}x$, $0 \leq x \leq h$

$V = \pi \int_0^h \left(\frac{R}{h}x\right)^2 dx = \frac{\pi R^2}{h^2} \int_0^h x^2 dx$

$= \frac{\pi R^2}{h^2} \cdot \frac{h^3}{3} = \frac{\pi R^2 h}{3}$

Javob: $V = \frac{1}{3}\pi R^2 h$

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7-bo'lim: Fizik masalalar (6 ta masala)

Masala 7.1 ⭐⭐⭐

Robot tezligi: $v(t) = 3t^2 - 2t$ m/s. $t = 0$ dan $t = 3s$ gacha bosib o'tgan masofa.

Yechim

$s = \int_0^3 |v(t)| dt$

$v(t) = t(3t - 2) = 0$ → $t = 0$ va $t = 2/3$

$[0, 2/3]$: $v \leq 0$, $[2/3, 3]$: $v \geq 0$

$s = -\int_0^{2/3} (3t^2 - 2t) dt + \int_{2/3}^3 (3t^2 - 2t) dt$

$= -[t^3 - t^2]_0^{2/3} + [t^3 - t^2]_{2/3}^3$

$= -(\frac{8}{27} - \frac{4}{9}) + (27 - 9 - \frac{8}{27} + \frac{4}{9})$

$= \frac{4}{27} + 18 - \frac{4}{27} = 18$ m

Javob: 18 m

Masala 7.2 ⭐⭐⭐

Raketa tezlanishi: $a(t) = 20 - 2t$ m/s². Boshlang'ich tezlik $v_0 = 0$. $t = 5s$ dagi tezlik va siljish.

Yechim

Tezlik: $v(t) = \int a(t) dt = 20t - t^2 + C$

$v(0) = 0$ → $C = 0$

$v(5) = 100 - 25 = 75$ m/s

Siljish: $s = \int_0^5 (20t - t^2) dt = [10t^2 - \frac{t^3}{3}]_0^5$

$= 250 - \frac{125}{3} = \frac{625}{3} \approx 208.3$ m

Javob: $v = 75$ m/s, $s \approx 208.3$ m

Masala 7.3 ⭐⭐⭐

Prujina kuchi: $F(x) = kx$, $k = 100$ N/m. Prujinani $x = 0$ dan $x = 0.2$ m ga cho'zish uchun ish.

Yechim

$W = \int_0^{0.2} F(x) dx = \int_0^{0.2} 100x dx$

$= [50x^2]_0^{0.2} = 50 \cdot 0.04 = 2$ J

Javob: $W = 2$ J

Masala 7.4 ⭐⭐⭐⭐

Quadcopter quvvat sarfi: $P(t) = 200 + 50\sin(t)$ W. 10 sekund davomida sarflangan energiya.

Yechim

$E = \int_0^{10} P(t) dt = \int_0^{10} (200 + 50\sin t) dt$

$= [200t - 50\cos t]_0^{10}$

$= (2000 - 50\cos 10) - (0 - 50)$

$= 2050 - 50\cos 10 \approx 2050 - 50(-0.839) \approx 2092$ J

Javob: $E \approx 2092$ J $\approx 2.1$ kJ

Masala 7.5 ⭐⭐⭐⭐

Raketa yoqilg'i sarfi: $\dot{m}(t) = 100e^{-0.1t}$ kg/s. 10 sekund davomida sarflangan yoqilg'i.

Yechim

$m = \int_0^{10} 100e^{-0.1t} dt = [-1000e^{-0.1t}]_0^{10}$

$= -1000e^{-1} + 1000 = 1000(1 - e^{-1})$

$\approx 1000(1 - 0.368) = 632$ kg

Javob: $m \approx 632$ kg

Masala 7.6 ⭐⭐⭐⭐

PID kontroller integral termi: $e(t) = 5e^{-0.5t}$, $K_i = 2$. $t = 0$ dan $t = 4s$ gacha integral qiymati.

Yechim

$u_I = K_i \int_0^4 e(t) dt = 2 \int_0^4 5e^{-0.5t} dt$

$= 10 \int_0^4 e^{-0.5t} dt = 10 \cdot [-2e^{-0.5t}]_0^4$

$= -20[e^{-2} - 1] = 20(1 - e^{-2})$

$\approx 20(1 - 0.135) = 17.3$

Javob: $u_I \approx 17.3$

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8-bo'lim: Noto'g'ri integrallar (3 ta masala)

Masala 8.1 ⭐⭐⭐

$\int_1^{\infty} \frac{1}{x^2} dx$

Yechim

$\lim_{b \to \infty} \int_1^b x^{-2} dx = \lim_{b \to \infty} [-x^{-1}]_1^b$

$= \lim_{b \to \infty} (-\frac{1}{b} + 1) = 0 + 1 = 1$

Javob: Integral yaqinlashadi, $= 1$

Masala 8.2 ⭐⭐⭐

$\int_0^{\infty} e^{-x} dx$

Yechim

$\lim_{b \to \infty} \int_0^b e^{-x} dx = \lim_{b \to \infty} [-e^{-x}]_0^b$

$= \lim_{b \to \infty} (-e^{-b} + 1) = 0 + 1 = 1$

Javob: $1$

Masala 8.3 ⭐⭐⭐⭐

$\int_0^1 \frac{1}{\sqrt{x}} dx$

Yechim

$x = 0$ da uzilish mavjud.

$\lim_{\epsilon \to 0^+} \int_{\epsilon}^1 x^{-1/2} dx = \lim_{\epsilon \to 0^+} [2\sqrt{x}]_{\epsilon}^1$

$= \lim_{\epsilon \to 0^+} (2 - 2\sqrt{\epsilon}) = 2 - 0 = 2$

Javob: $2$

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9-bo'lim: Raqamli integrallash (2 ta masala)

Masala 9.1 ⭐⭐⭐

Trapetsiya qoidasi bilan hisoblang ($n = 4$): $\int_0^2 e^{-x^2} dx$

Yechim

$h = \frac{2-0}{4} = 0.5$

$x_i$: 0, 0.5, 1.0, 1.5, 2.0

$f(x_i)$: 1, 0.779, 0.368, 0.105, 0.018

$I \approx \frac{0.5}{2}[f(0) + 2f(0.5) + 2f(1) + 2f(1.5) + f(2)]$

$= 0.25[1 + 1.558 + 0.736 + 0.210 + 0.018]$

$= 0.25 \times 3.522 = 0.881$

Javob: $\approx 0.881$ (aniq qiymat: $\approx 0.882$)

Masala 9.2 ⭐⭐⭐⭐

Simpson qoidasi bilan hisoblang ($n = 4$): $\int_0^1 \sin(\pi x) dx$

Yechim

$h = 0.25$

$x_i$: 0, 0.25, 0.5, 0.75, 1.0

$f(x_i)$: 0, 0.707, 1, 0.707, 0

Simpson: $I \approx \frac{h}{3}[f_0 + 4f_1 + 2f_2 + 4f_3 + f_4]$

$= \frac{0.25}{3}[0 + 4(0.707) + 2(1) + 4(0.707) + 0]$

$= \frac{0.25}{3}[5.656 + 2] = \frac{0.25 \times 7.656}{3} = 0.638$

Aniq qiymat: $\frac{2}{\pi} = 0.637$

Javob: $\approx 0.638$

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Qiyinlik darajasi bo'yicha

Daraja	Masalalar soni
⭐⭐ Oson	9
⭐⭐⭐ O'rta	16
⭐⭐⭐⭐ Murakkab	10
Jami	35