Integral kalkulyatori

Ushbu kalkulyatordan foydalanib, siz aniqlanmagan yoki aniq integralni hisoblashingiz mumkin. Hisoblash misollari tegishli bo’limda topilishi mumkin.

Ba’zi asosiy integralllarni hisoblash o’rniga, birlamchi birinchi funksiyani jadvaldoq topishingiz mumkin.

Birinchi darajali

Funksiya $f(x)$ uchun birinchi darajali $F(x)$ - bu o’zining hosilasi $f(x)$ ga teng bo’lgan funksiya, ya’ni $F^{\prime}(x) = f(x)$. Birinchi darajali topish differensiallash jarayonining teskarisidir.

Agar $F(x)$ funksiya $f(x)$ uchun birinchi darajali bo’lsa, $F(x) + C$ funksiyasi ham $f(x)$ ning birinchi darajalisi hisoblanadi, bu yerda $C$ - ixtiyoriy doimiy.

Aniqlanmagan integral

Funksiya $f(x)$ uchun aniqlanmagan integral - bu funksiyaning barcha birinchi darajalilari jamlanmasidan iborat. Bu quyidagicha ifodalanadi:

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

bu yerda

$\int$ - integral belgisi
$f(x)$ - integralga olinadigan funksiya
$dx$ - integrallash elementi
$F(x)$ - birinchi darajali
$C$ - integrallash doimiy miqdori

Integral topish jarayoni integrallash deb ataladi.

Xossalari

Aniqlanmagan integralning asosiy xossalari:

$$\int a \cdot f(x) dx = \\ = a \cdot \int f(x) dx \ \ \ (a \ne 0)$$

$$\int (f(x) \pm g(x)) dx = \\ = \int f(x) dx \pm \int g(x) dx$$

Hisoblash misollari

Quyida aniqlanmagan integralllarni hisoblash misollari keltirilgan. Integralni kalkulyatorda hisoblash uchun har bir misoldan keyin ko’rsatilgan tugmalarni bosish kerak. Eslatma: integral kalkulyatorining ekranidagi bo’sh maydonchaga kompyuter klaviaturasini ishlatib, int deb kiriting.

$$\int x^3 dx = \frac {x^4} {4} + C$$

i n t ( x x^y 3 ) =

$$\int \sin 7x dx = - \frac {\cos 7x} {7} + C$$

i n t ( sin 7 x ) =

$$\int x^3 y dy = \frac {x^3 y^2} {2} + C$$

i n t ( x x^3 y , y ) =

Integralllar jadvali

Asosiy aniqlanmagan integralllar va ularning birinchi darajalilarining jadvali:

$\int f(x) dx$	$F(x) + C$
$$\int 0 \cdot dx$$	$$C$$
$$\int dx$$	$$x + C$$
$$\int x^n dx$$	$$\frac {x^{n + 1}} {n + 1} + C \ \ \ (n \ne -1)$$
$$\int \frac {1} {x} dx$$	$$\ln \| x \| + C$$
$$\int e^x dx$$	$$e^x + C$$
$$\int a^x dx$$	$$\frac {a^x} {\ln a} + C \ \ \ (a > 0, a \ne 1)$$
$$\int \cos x dx$$	$$\sin x + C$$
$$\int \sin x dx$$	$$- \cos x + C$$
$$\int \frac {dx} {\cos^2 x}$$	$$\tg x + C$$
$$\int \frac {dx} {\sin^2 x}$$	$$- \ctg x + C$$
$$\int \frac {dx} {\sqrt {1 - x^2}}$$	$$\begin{gathered} \arcsin x + C_1 = \\ = - \arccos x + C_2 \\ (C_2 = \frac {\pi} {2} + C_1) \end{gathered}$$
$$\int \frac {dx} {1 + x^2}$$	$$\arctg x + C$$
$$\int \ch x dx$$	$$\sh x + C$$
$$\int \sh x dx$$	$$\ch x + C$$

Aniq integral

Agar $F(x)$ funksiyasi $f(x)$ uchun birinchi darajali bo’lib, u $[a;b]$ bo’lagida aniqlanib va uzluksiz bo’lsa, aniq integral quyidagi formula bo’yicha hisoblanadi:

$$\int _a^b f(x) dx = F(x) \mid _a^b = F(b) - F(a)$$

Xossalari

Aniq integralning asosiy xossalari:

$$\int _a^a f(x) dx = 0$$

$$\int _a^b f(x) dx = - \int _b^a f(x) dx$$

$$\int _a^b f(x) dx = \int _a^c f(x) dx + \\ + \int _c^b f(x) dx \ \ \ (c \in [a;b])$$

Hisoblash misollari

Quyida aniq integralllarni hisoblash misollari keltirilgan. Ularni kalkulyatorda hisoblash uchun har bir misoldan keyin ko’rsatilgan tugmalarni bosish kerak. Eslatma: integral kalkulyatorining ekranidagi bo’sh maydonchaga kompyuter klaviaturasini ishlatib, int deb kiriting.

$$\int _1^3 (5 + x) dx = 14$$

i n t ( 5 + x ,
x 2^nd var 1 . . 3 2^nd ) =

$$\int _5^8 x^2 dx = 129$$

i n t ( x x² ,
x 2^nd var 5 . . 8 2^nd ) =