Jadual Integral Komprehensif
Panduan rujukan lengkap untuk integral tak tentu - lebih 100 formula
Integral Asas
| Fungsi f(x) | Integral ∫f(x)dx | Domain |
|---|---|---|
| \(k\) (pemalar) | \(kx + C\) | Semua nombor nyata |
| \(x^n\) (di mana \(n ≠ -1\)) | \(\displaystyle\frac{x^{n+1}}{n+1} + C\) | \(x ≥ 0\) jika \(n < 0\), selain itu semua nombor nyata |
| \(\displaystyle\frac{1}{x}\) | \(\ln|x| + C\) | \(x ≠ 0\) |
| \(\sqrt{x}\) | \(\displaystyle\frac{2x^{3/2}}{3} + C\) | \(x ≥ 0\) |
| \(\displaystyle\frac{1}{\sqrt{x}}\) | \(2\sqrt{x} + C\) | \(x > 0\) |
| \(\displaystyle\frac{1}{x^2}\) | \(-\displaystyle\frac{1}{x} + C\) | \(x ≠ 0\) |
| \(\displaystyle\frac{1}{x^3}\) | \(-\displaystyle\frac{1}{2x^2} + C\) | \(x ≠ 0\) |
| \(x^{1/2}\) | \(\displaystyle\frac{2x^{3/2}}{3} + C\) | \(x ≥ 0\) |
| \(x^{-1/2}\) | \(2x^{1/2} + C\) | \(x > 0\) |
| \(x^{1/3}\) | \(\displaystyle\frac{3x^{4/3}}{4} + C\) | Semua nombor nyata |
Fungsi Eksponen dan Logaritma
| Fungsi f(x) | Integral ∫f(x)dx | Domain |
|---|---|---|
| \(e^x\) | \(e^x + C\) | Semua nombor nyata |
| \(a^x\) (di mana \(a > 0, a ≠ 1\)) | \(\displaystyle\frac{a^x}{\ln a} + C\) | Semua nombor nyata |
| \(e^{ax}\) (di mana \(a ≠ 0\)) | \(\displaystyle\frac{e^{ax}}{a} + C\) | Semua nombor nyata |
| \(\ln x\) | \(x \ln x - x + C\) | \(x > 0\) |
| \(\log_a x\) (di mana \(a > 0, a ≠ 1\)) | \(\displaystyle\frac{x \ln x - x}{\ln a} + C\) | \(x > 0\) |
| \(xe^x\) | \((x-1)e^x + C\) | Semua nombor nyata |
| \(x^2e^x\) | \((x^2-2x+2)e^x + C\) | Semua nombor nyata |
| \(e^{ax}\sin(bx)\) | \(\displaystyle\frac{e^{ax}(a\sin(bx) - b\cos(bx))}{a^2 + b^2} + C\) | Semua nombor nyata |
| \(e^{ax}\cos(bx)\) | \(\displaystyle\frac{e^{ax}(a\cos(bx) + b\sin(bx))}{a^2 + b^2} + C\) | Semua nombor nyata |
| \(\displaystyle\frac{\ln x}{x}\) | \(\displaystyle\frac{(\ln x)^2}{2} + C\) | \(x > 0\) |
| \((\ln x)^n\) | \(x(\ln x)^n - n\int (\ln x)^{n-1} dx\) | \(x > 0\) |
| \(e^{-x^2}\) | \(\displaystyle\frac{\sqrt{\pi}}{2}\text{erf}(x) + C\) | Semua nombor nyata |
Fungsi Trigonometri
| Fungsi f(x) | Integral ∫f(x)dx | Domain |
|---|---|---|
| \(\sin x\) | \(-\cos x + C\) | Semua nombor nyata |
| \(\cos x\) | \(\sin x + C\) | Semua nombor nyata |
| \(\tan x\) | \(-\ln|\cos x| + C\) | \(x ≠ \frac{\pi}{2} + \pi n\) |
| \(\cot x\) | \(\ln|\sin x| + C\) | \(x ≠ \pi n\) |
| \(\sec x\) | \(\ln|\sec x + \tan x| + C\) | \(x ≠ \frac{\pi}{2} + \pi n\) |
| \(\csc x\) | \(-\ln|\csc x + \cot x| + C\) | \(x ≠ \pi n\) |
| \(\sec^2 x\) | \(\tan x + C\) | \(x ≠ \frac{\pi}{2} + \pi n\) |
| \(\csc^2 x\) | \(-\cot x + C\) | \(x ≠ \pi n\) |
| \(\sec x \tan x\) | \(\sec x + C\) | \(x ≠ \frac{\pi}{2} + \pi n\) |
| \(\csc x \cot x\) | \(-\csc x + C\) | \(x ≠ \pi n\) |
| \(\sin^2 x\) | \(\displaystyle\frac{x}{2} - \frac{\sin 2x}{4} + C\) | Semua nombor nyata |
| \(\cos^2 x\) | \(\displaystyle\frac{x}{2} + \frac{\sin 2x}{4} + C\) | Semua nombor nyata |
| \(\sin x \cos x\) | \(\displaystyle\frac{\sin^2 x}{2} + C\) | Semua nombor nyata |
| \(\sin^3 x\) | \(-\cos x + \displaystyle\frac{\cos^3 x}{3} + C\) | Semua nombor nyata |
| \(\cos^3 x\) | \(\sin x - \displaystyle\frac{\sin^3 x}{3} + C\) | Semua nombor nyata |
| \(\tan^2 x\) | \(\tan x - x + C\) | \(x ≠ \frac{\pi}{2} + \pi n\) |
| \(\sin(ax)\) | \(-\displaystyle\frac{\cos(ax)}{a} + C\) | Semua nombor nyata, \(a ≠ 0\) |
| \(\cos(ax)\) | \(\displaystyle\frac{\sin(ax)}{a} + C\) | Semua nombor nyata, \(a ≠ 0\) |
| \(\sin^n x\) | \(-\displaystyle\frac{\sin^{n-1} x \cos x}{n} + \frac{n-1}{n}\int \sin^{n-2} x dx\) | Semua nombor nyata |
| \(\cos^n x\) | \(\displaystyle\frac{\cos^{n-1} x \sin x}{n} + \frac{n-1}{n}\int \cos^{n-2} x dx\) | Semua nombor nyata |
Fungsi Trigonometri Songsang
| Fungsi f(x) | Integral ∫f(x)dx | Domain |
|---|---|---|
| \(\displaystyle\frac{1}{\sqrt{1-x^2}}\) | \(\arcsin x + C\) | \(-1 < x < 1\) |
| \(\displaystyle-\frac{1}{\sqrt{1-x^2}}\) | \(\arccos x + C\) | \(-1 < x < 1\) |
| \(\displaystyle\frac{1}{1+x^2}\) | \(\arctan x + C\) | Semua nombor nyata |
| \(\displaystyle-\frac{1}{1+x^2}\) | \(\text{arccot } x + C\) | Semua nombor nyata |
| \(\displaystyle\frac{1}{|x|\sqrt{x^2-1}}\) | \(\text{arcsec } |x| + C\) | \(|x| > 1\) |
| \(\arcsin x\) | \(x \arcsin x + \sqrt{1-x^2} + C\) | \(-1 ≤ x ≤ 1\) |
| \(\arccos x\) | \(x \arccos x - \sqrt{1-x^2} + C\) | \(-1 ≤ x ≤ 1\) |
| \(\arctan x\) | \(x \arctan x - \displaystyle\frac{1}{2}\ln(1+x^2) + C\) | Semua nombor nyata |
| \(\displaystyle\frac{1}{\sqrt{a^2-x^2}}\) | \(\arcsin\displaystyle\frac{x}{a} + C\) | \(|x| < a\) |
| \(\displaystyle\frac{1}{a^2+x^2}\) | \(\displaystyle\frac{1}{a}\arctan\frac{x}{a} + C\) | Semua nombor nyata, \(a ≠ 0\) |
Fungsi Hiperbolik
| Fungsi f(x) | Integral ∫f(x)dx | Domain |
|---|---|---|
| \(\sinh x\) | \(\cosh x + C\) | Semua nombor nyata |
| \(\cosh x\) | \(\sinh x + C\) | Semua nombor nyata |
| \(\tanh x\) | \(\ln(\cosh x) + C\) | Semua nombor nyata |
| \(\coth x\) | \(\ln|\sinh x| + C\) | \(x ≠ 0\) |
| \(\text{sech}^2 x\) | \(\tanh x + C\) | Semua nombor nyata |
| \(\text{csch}^2 x\) | \(-\coth x + C\) | \(x ≠ 0\) |
| \(\sinh^2 x\) | \(\displaystyle\frac{\sinh 2x}{4} - \frac{x}{2} + C\) | Semua nombor nyata |
| \(\cosh^2 x\) | \(\displaystyle\frac{\sinh 2x}{4} + \frac{x}{2} + C\) | Semua nombor nyata |
| \(\text{sech } x\) | \(\arctan(\sinh x) + C\) | Semua nombor nyata |
| \(\text{csch } x\) | \(\ln\left|\tanh\displaystyle\frac{x}{2}\right| + C\) | \(x ≠ 0\) |
Fungsi Rasional
| Fungsi f(x) | Integral ∫f(x)dx | Domain |
|---|---|---|
| \(\displaystyle\frac{1}{ax + b}\) | \(\displaystyle\frac{\ln|ax + b|}{a} + C\) | \(x ≠ -\frac{b}{a}, a ≠ 0\) |
| \(\displaystyle\frac{1}{(ax + b)^2}\) | \(-\displaystyle\frac{1}{a(ax + b)} + C\) | \(x ≠ -\frac{b}{a}, a ≠ 0\) |
| \(\displaystyle\frac{1}{x^2 + a^2}\) | \(\displaystyle\frac{1}{a} \arctan\frac{x}{a} + C\) | Semua nombor nyata, \(a ≠ 0\) |
| \(\displaystyle\frac{1}{x^2 - a^2}\) | \(\displaystyle\frac{1}{2a} \ln\left|\frac{x-a}{x+a}\right| + C\) | \(x ≠ ±a, a ≠ 0\) |
| \(\displaystyle\frac{1}{a^2 - x^2}\) | \(\displaystyle\frac{1}{2a} \ln\left|\frac{a+x}{a-x}\right| + C\) | \(x ≠ ±a, a ≠ 0\) |
| \(\displaystyle\frac{x}{x^2 + a^2}\) | \(\displaystyle\frac{1}{2}\ln(x^2 + a^2) + C\) | Semua nombor nyata |
| \(\displaystyle\frac{x}{x^2 - a^2}\) | \(\displaystyle\frac{1}{2}\ln|x^2 - a^2| + C\) | \(x ≠ ±a\) |
| \(\displaystyle\frac{1}{(x^2 + a^2)^2}\) | \(\displaystyle\frac{x}{2a^2(x^2 + a^2)} + \frac{1}{2a^3}\arctan\frac{x}{a} + C\) | Semua nombor nyata, \(a ≠ 0\) |
| \(\displaystyle\frac{x^2}{x^2 + a^2}\) | \(x - a\arctan\displaystyle\frac{x}{a} + C\) | Semua nombor nyata, \(a ≠ 0\) |
| \(\displaystyle\frac{1}{x(x+a)}\) | \(\displaystyle\frac{1}{a}\ln\left|\frac{x}{x+a}\right| + C\) | \(x ≠ 0, -a\) |
Fungsi Radikal
| Fungsi f(x) | Integral ∫f(x)dx | Domain |
|---|---|---|
| \(\displaystyle\frac{1}{\sqrt{x^2 + a^2}}\) | \(\ln(x + \sqrt{x^2 + a^2}) + C\) | Semua nombor nyata |
| \(\displaystyle\frac{1}{\sqrt{x^2 - a^2}}\) | \(\ln|x + \sqrt{x^2 - a^2}| + C\) | \(|x| > a\) |
| \(\displaystyle\frac{1}{\sqrt{a^2 - x^2}}\) | \(\arcsin\displaystyle\frac{x}{a} + C\) | \(|x| < a\) |
| \(\sqrt{a^2 - x^2}\) | \(\displaystyle\frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\arcsin\frac{x}{a} + C\) | \(-a ≤ x ≤ a\) |
| \(\sqrt{x^2 + a^2}\) | \(\displaystyle\frac{x}{2}\sqrt{x^2+a^2} + \frac{a^2}{2}\ln(x + \sqrt{x^2+a^2}) + C\) | Semua nombor nyata |
| \(\sqrt{x^2 - a^2}\) | \(\displaystyle\frac{x}{2}\sqrt{x^2-a^2} - \frac{a^2}{2}\ln|x + \sqrt{x^2-a^2}| + C\) | \(|x| ≥ a\) |
| \(\displaystyle\frac{x}{\sqrt{x^2 + a^2}}\) | \(\sqrt{x^2 + a^2} + C\) | Semua nombor nyata |
| \(\displaystyle\frac{x}{\sqrt{x^2 - a^2}}\) | \(\sqrt{x^2 - a^2} + C\) | \(|x| > a\) |
| \(\displaystyle\frac{x}{\sqrt{a^2 - x^2}}\) | \(-\sqrt{a^2 - x^2} + C\) | \(|x| < a\) |
| \(\sqrt{ax + b}\) | \(\displaystyle\frac{2(ax + b)^{3/2}}{3a} + C\) | \(ax + b ≥ 0, a ≠ 0\) |
| \(\displaystyle\frac{1}{\sqrt{ax + b}}\) | \(\displaystyle\frac{2\sqrt{ax + b}}{a} + C\) | \(ax + b > 0, a ≠ 0\) |
| \(x\sqrt{x^2 + a^2}\) | \(\displaystyle\frac{(x^2 + a^2)^{3/2}}{3} + C\) | Semua nombor nyata |
Hasil Darab dengan x
| Fungsi f(x) | Integral ∫f(x)dx | Domain |
|---|---|---|
| \(x\sin x\) | \(\sin x - x\cos x + C\) | Semua nombor nyata |
| \(x\cos x\) | \(\cos x + x\sin x + C\) | Semua nombor nyata |
| \(x^2\sin x\) | \((2 - x^2)\cos x + 2x\sin x + C\) | Semua nombor nyata |
| \(x^2\cos x\) | \((x^2 - 2)\sin x + 2x\cos x + C\) | Semua nombor nyata |
| \(x \ln x\) | \(\displaystyle\frac{x^2 \ln x}{2} - \frac{x^2}{4} + C\) | \(x > 0\) |
| \(x^2 \ln x\) | \(\displaystyle\frac{x^3 \ln x}{3} - \frac{x^3}{9} + C\) | \(x > 0\) |
| \(x^n \ln x\) | \(\displaystyle\frac{x^{n+1} \ln x}{n+1} - \frac{x^{n+1}}{(n+1)^2} + C\) | \(x > 0, n ≠ -1\) |
| \(x \arcsin x\) | \(\displaystyle\frac{x^2 \arcsin x}{2} + \frac{\sqrt{1-x^2}}{2} - \frac{x}{2} + C\) | \(-1 ≤ x ≤ 1\) |
| \(x \arctan x\) | \(\displaystyle\frac{x^2 \arctan x}{2} - \frac{x}{2} + \frac{\arctan x}{2} + C\) | Semua nombor nyata |
| \(x \sinh x\) | \(x \cosh x - \sinh x + C\) | Semua nombor nyata |
| \(x \cosh x\) | \(x \sinh x - \cosh x + C\) | Semua nombor nyata |
| \(x^n e^{ax}\) | \(\displaystyle\frac{x^n e^{ax}}{a} - \frac{n}{a}\int x^{n-1} e^{ax} dx\) | Semua nombor nyata, \(a ≠ 0\) |
| \(x^3 e^x\) | \((x^3 - 3x^2 + 6x - 6)e^x + C\) | Semua nombor nyata |
| \(x\tan x\) | \(x\ln|\cos x| + \displaystyle\frac{x^2}{2} + C\) | \(x ≠ \frac{\pi}{2} + \pi n\) |
Peraturan Asas Pengamiran
Kelinearan:
\(\displaystyle\int [af(x) + bg(x)] dx = a\int f(x) dx + b\int g(x) dx\)
\(\displaystyle\int [af(x) + bg(x)] dx = a\int f(x) dx + b\int g(x) dx\)
Pengamiran mengikut bahagian:
\(\displaystyle\int u \, dv = uv - \int v \, du\)
\(\displaystyle\int u \, dv = uv - \int v \, du\)
Penggantian:
\(\displaystyle\int f(\varphi(x))\varphi'(x) dx = \int f(u) du\), di mana \(u = \varphi(x)\)
\(\displaystyle\int f(\varphi(x))\varphi'(x) dx = \int f(u) du\), di mana \(u = \varphi(x)\)
Fungsi Genap:
\(\displaystyle\int_{-a}^{a} f(x) dx = 2\int_{0}^{a} f(x) dx\) (jika \(f(-x) = f(x)\))
\(\displaystyle\int_{-a}^{a} f(x) dx = 2\int_{0}^{a} f(x) dx\) (jika \(f(-x) = f(x)\))
Fungsi Ganjil:
\(\displaystyle\int_{-a}^{a} f(x) dx = 0\) (jika \(f(-x) = -f(x)\))
\(\displaystyle\int_{-a}^{a} f(x) dx = 0\) (jika \(f(-x) = -f(x)\))
Teorem Asas Kalkulus:
\(\displaystyle\int_a^b f(x) dx = F(b) - F(a)\), di mana \(F'(x) = f(x)\)
\(\displaystyle\int_a^b f(x) dx = F(b) - F(a)\), di mana \(F'(x) = f(x)\)
Tiada hasil ditemui
Cuba ubah suai pertanyaan carian anda atau kosongkan medan carian
Interpretasi Geometri Pengamiran
Rujukan Integrasi Matematik Komprehensif
Jadual kamiran ini menyediakan panduan rujukan lengkap untuk kamiran tak tentu fungsi matematik biasa. Jadual ini menyusun formula kamiran mengikut jenis fungsi, termasuk polinomial asas, fungsi eksponen dan logaritma, fungsi trigonometri, fungsi trigonometri songsang, dan ungkapan lanjutan yang melibatkan radikal dan fungsi rasional.
Formula Integrasi Am:
∫ f(x)dx = F(x) + C
Di mana F(x) ialah antiterbitan f(x) dan C ialah pemalar integrasi.
∫ f(x)dx = F(x) + C
Di mana F(x) ialah antiterbitan f(x) dan C ialah pemalar integrasi.
Setiap entri dalam jadual rujukan ini termasuk fungsi asal f(x), kamiran sepadannya ∫f(x)dx, dan domain kesahihan. Jadual ini meliputi peraturan integrasi asas, aplikasi peraturan kuasa, identiti trigonometri, integrasi logaritma dan eksponen, serta teknik lanjutan untuk ungkapan radikal.
Kategori Fungsi yang Diliputi
Fungsi Asas
Pemalar, kuasa xn, salingan 1/x, punca kuasa dua √x
Eksponen & Logaritma
ex, ax, ln(x), loga(x), xex
Trigonometri
sin(x), cos(x), tan(x), sec2(x), sin2(x), cos2(x)
Trigonometri Songsang
1/√(1-x2), 1/(1+x2), arcsin(x), arctan(x)
Ungkapan Lanjutan
√(a2-x2), 1/√(x2±a2), fungsi rasional
Contoh Penggunaan Jadual Rujukan
- Mencari ∫x3dx = x4/4 + C untuk integrasi polinomial
- Mencari ∫e2xdx = e2x/2 + C untuk fungsi eksponen
- Merujuk ∫sin(x)dx = -cos(x) + C untuk integrasi trigonometri
- Memeriksa ∫1/√(1-x2)dx = arcsin(x) + C untuk fungsi trigonometri songsang
- Mencari ∫ln(x)dx = x ln(x) - x + C untuk fungsi logaritma
- Mencari ∫1/(x2+4)dx = (1/2)arctan(x/2) + C untuk ungkapan rasional
- Mencari ∫√(9-x2)dx untuk formula integrasi radikal
- Merujuk ∫x·cos(x)dx = cos(x) + x·sin(x) + C untuk hasil darab
- Memeriksa ∫sec2(x)dx = tan(x) + C untuk fungsi sekan
- Mencari ∫1/√(x2+1)dx = ln(x + √(x2+1)) + C untuk bentuk hiperbolik
Rujukan ini termasuk had domain untuk setiap kamiran, peraturan integrasi seperti kelinearan dan kaedah penggantian, serta representasi visual yang menunjukkan integrasi sebagai luas di bawah lengkung. Fungsi carian membolehkan pencarian pantas jenis fungsi atau ungkapan matematik tertentu.