Comprehensive Integrals Table
Complete reference guide for indefinite integrals - over 100 formulas
Basic Integrals
| Function f(x) | Integral ∫f(x)dx | Domain |
|---|---|---|
| \(k\) (constant) | \(kx + C\) | All real numbers |
| \(x^n\) (where \(n ≠ -1\)) | \(\displaystyle\frac{x^{n+1}}{n+1} + C\) | \(x ≥ 0\) if \(n < 0\), otherwise all real numbers |
| \(\displaystyle\frac{1}{x}\) | \(\ln|x| + C\) | \(x ≠ 0\) |
| \(\sqrt{x}\) | \(\displaystyle\frac{2x^{3/2}}{3} + C\) | \(x ≥ 0\) |
| \(\displaystyle\frac{1}{\sqrt{x}}\) | \(2\sqrt{x} + C\) | \(x > 0\) |
| \(\displaystyle\frac{1}{x^2}\) | \(-\displaystyle\frac{1}{x} + C\) | \(x ≠ 0\) |
| \(\displaystyle\frac{1}{x^3}\) | \(-\displaystyle\frac{1}{2x^2} + C\) | \(x ≠ 0\) |
| \(x^{1/2}\) | \(\displaystyle\frac{2x^{3/2}}{3} + C\) | \(x ≥ 0\) |
| \(x^{-1/2}\) | \(2x^{1/2} + C\) | \(x > 0\) |
| \(x^{1/3}\) | \(\displaystyle\frac{3x^{4/3}}{4} + C\) | All real numbers |
Exponential and Logarithmic Functions
| Function f(x) | Integral ∫f(x)dx | Domain |
|---|---|---|
| \(e^x\) | \(e^x + C\) | All real numbers |
| \(a^x\) (where \(a > 0, a ≠ 1\)) | \(\displaystyle\frac{a^x}{\ln a} + C\) | All real numbers |
| \(e^{ax}\) (where \(a ≠ 0\)) | \(\displaystyle\frac{e^{ax}}{a} + C\) | All real numbers |
| \(\ln x\) | \(x \ln x - x + C\) | \(x > 0\) |
| \(\log_a x\) (where \(a > 0, a ≠ 1\)) | \(\displaystyle\frac{x \ln x - x}{\ln a} + C\) | \(x > 0\) |
| \(xe^x\) | \((x-1)e^x + C\) | All real numbers |
| \(x^2e^x\) | \((x^2-2x+2)e^x + C\) | All real numbers |
| \(e^{ax}\sin(bx)\) | \(\displaystyle\frac{e^{ax}(a\sin(bx) - b\cos(bx))}{a^2 + b^2} + C\) | All real numbers |
| \(e^{ax}\cos(bx)\) | \(\displaystyle\frac{e^{ax}(a\cos(bx) + b\sin(bx))}{a^2 + b^2} + C\) | All real numbers |
| \(\displaystyle\frac{\ln x}{x}\) | \(\displaystyle\frac{(\ln x)^2}{2} + C\) | \(x > 0\) |
| \((\ln x)^n\) | \(x(\ln x)^n - n\int (\ln x)^{n-1} dx\) | \(x > 0\) |
| \(e^{-x^2}\) | \(\displaystyle\frac{\sqrt{\pi}}{2}\text{erf}(x) + C\) | All real numbers |
Trigonometric Functions
| Function f(x) | Integral ∫f(x)dx | Domain |
|---|---|---|
| \(\sin x\) | \(-\cos x + C\) | All real numbers |
| \(\cos x\) | \(\sin x + C\) | All real numbers |
| \(\tan x\) | \(-\ln|\cos x| + C\) | \(x ≠ \frac{\pi}{2} + \pi n\) |
| \(\cot x\) | \(\ln|\sin x| + C\) | \(x ≠ \pi n\) |
| \(\sec x\) | \(\ln|\sec x + \tan x| + C\) | \(x ≠ \frac{\pi}{2} + \pi n\) |
| \(\csc x\) | \(-\ln|\csc x + \cot x| + C\) | \(x ≠ \pi n\) |
| \(\sec^2 x\) | \(\tan x + C\) | \(x ≠ \frac{\pi}{2} + \pi n\) |
| \(\csc^2 x\) | \(-\cot x + C\) | \(x ≠ \pi n\) |
| \(\sec x \tan x\) | \(\sec x + C\) | \(x ≠ \frac{\pi}{2} + \pi n\) |
| \(\csc x \cot x\) | \(-\csc x + C\) | \(x ≠ \pi n\) |
| \(\sin^2 x\) | \(\displaystyle\frac{x}{2} - \frac{\sin 2x}{4} + C\) | All real numbers |
| \(\cos^2 x\) | \(\displaystyle\frac{x}{2} + \frac{\sin 2x}{4} + C\) | All real numbers |
| \(\sin x \cos x\) | \(\displaystyle\frac{\sin^2 x}{2} + C\) | All real numbers |
| \(\sin^3 x\) | \(-\cos x + \displaystyle\frac{\cos^3 x}{3} + C\) | All real numbers |
| \(\cos^3 x\) | \(\sin x - \displaystyle\frac{\sin^3 x}{3} + C\) | All real numbers |
| \(\tan^2 x\) | \(\tan x - x + C\) | \(x ≠ \frac{\pi}{2} + \pi n\) |
| \(\sin(ax)\) | \(-\displaystyle\frac{\cos(ax)}{a} + C\) | All real numbers, \(a ≠ 0\) |
| \(\cos(ax)\) | \(\displaystyle\frac{\sin(ax)}{a} + C\) | All real numbers, \(a ≠ 0\) |
| \(\sin^n x\) | \(-\displaystyle\frac{\sin^{n-1} x \cos x}{n} + \frac{n-1}{n}\int \sin^{n-2} x dx\) | All real numbers |
| \(\cos^n x\) | \(\displaystyle\frac{\cos^{n-1} x \sin x}{n} + \frac{n-1}{n}\int \cos^{n-2} x dx\) | All real numbers |
Inverse Trigonometric Functions
| Function f(x) | Integral ∫f(x)dx | Domain |
|---|---|---|
| \(\displaystyle\frac{1}{\sqrt{1-x^2}}\) | \(\arcsin x + C\) | \(-1 < x < 1\) |
| \(\displaystyle-\frac{1}{\sqrt{1-x^2}}\) | \(\arccos x + C\) | \(-1 < x < 1\) |
| \(\displaystyle\frac{1}{1+x^2}\) | \(\arctan x + C\) | All real numbers |
| \(\displaystyle-\frac{1}{1+x^2}\) | \(\text{arccot } x + C\) | All real numbers |
| \(\displaystyle\frac{1}{|x|\sqrt{x^2-1}}\) | \(\text{arcsec } |x| + C\) | \(|x| > 1\) |
| \(\arcsin x\) | \(x \arcsin x + \sqrt{1-x^2} + C\) | \(-1 ≤ x ≤ 1\) |
| \(\arccos x\) | \(x \arccos x - \sqrt{1-x^2} + C\) | \(-1 ≤ x ≤ 1\) |
| \(\arctan x\) | \(x \arctan x - \displaystyle\frac{1}{2}\ln(1+x^2) + C\) | All real numbers |
| \(\displaystyle\frac{1}{\sqrt{a^2-x^2}}\) | \(\arcsin\displaystyle\frac{x}{a} + C\) | \(|x| < a\) |
| \(\displaystyle\frac{1}{a^2+x^2}\) | \(\displaystyle\frac{1}{a}\arctan\frac{x}{a} + C\) | All real numbers, \(a ≠ 0\) |
Hyperbolic Functions
| Function f(x) | Integral ∫f(x)dx | Domain |
|---|---|---|
| \(\sinh x\) | \(\cosh x + C\) | All real numbers |
| \(\cosh x\) | \(\sinh x + C\) | All real numbers |
| \(\tanh x\) | \(\ln(\cosh x) + C\) | All real numbers |
| \(\coth x\) | \(\ln|\sinh x| + C\) | \(x ≠ 0\) |
| \(\text{sech}^2 x\) | \(\tanh x + C\) | All real numbers |
| \(\text{csch}^2 x\) | \(-\coth x + C\) | \(x ≠ 0\) |
| \(\sinh^2 x\) | \(\displaystyle\frac{\sinh 2x}{4} - \frac{x}{2} + C\) | All real numbers |
| \(\cosh^2 x\) | \(\displaystyle\frac{\sinh 2x}{4} + \frac{x}{2} + C\) | All real numbers |
| \(\text{sech } x\) | \(\arctan(\sinh x) + C\) | All real numbers |
| \(\text{csch } x\) | \(\ln\left|\tanh\displaystyle\frac{x}{2}\right| + C\) | \(x ≠ 0\) |
Rational Functions
| Function f(x) | Integral ∫f(x)dx | Domain |
|---|---|---|
| \(\displaystyle\frac{1}{ax + b}\) | \(\displaystyle\frac{\ln|ax + b|}{a} + C\) | \(x ≠ -\frac{b}{a}, a ≠ 0\) |
| \(\displaystyle\frac{1}{(ax + b)^2}\) | \(-\displaystyle\frac{1}{a(ax + b)} + C\) | \(x ≠ -\frac{b}{a}, a ≠ 0\) |
| \(\displaystyle\frac{1}{x^2 + a^2}\) | \(\displaystyle\frac{1}{a} \arctan\frac{x}{a} + C\) | All real numbers, \(a ≠ 0\) |
| \(\displaystyle\frac{1}{x^2 - a^2}\) | \(\displaystyle\frac{1}{2a} \ln\left|\frac{x-a}{x+a}\right| + C\) | \(x ≠ ±a, a ≠ 0\) |
| \(\displaystyle\frac{1}{a^2 - x^2}\) | \(\displaystyle\frac{1}{2a} \ln\left|\frac{a+x}{a-x}\right| + C\) | \(x ≠ ±a, a ≠ 0\) |
| \(\displaystyle\frac{x}{x^2 + a^2}\) | \(\displaystyle\frac{1}{2}\ln(x^2 + a^2) + C\) | All real numbers |
| \(\displaystyle\frac{x}{x^2 - a^2}\) | \(\displaystyle\frac{1}{2}\ln|x^2 - a^2| + C\) | \(x ≠ ±a\) |
| \(\displaystyle\frac{1}{(x^2 + a^2)^2}\) | \(\displaystyle\frac{x}{2a^2(x^2 + a^2)} + \frac{1}{2a^3}\arctan\frac{x}{a} + C\) | All real numbers, \(a ≠ 0\) |
| \(\displaystyle\frac{x^2}{x^2 + a^2}\) | \(x - a\arctan\displaystyle\frac{x}{a} + C\) | All real numbers, \(a ≠ 0\) |
| \(\displaystyle\frac{1}{x(x+a)}\) | \(\displaystyle\frac{1}{a}\ln\left|\frac{x}{x+a}\right| + C\) | \(x ≠ 0, -a\) |
Radical Functions
| Function f(x) | Integral ∫f(x)dx | Domain |
|---|---|---|
| \(\displaystyle\frac{1}{\sqrt{x^2 + a^2}}\) | \(\ln(x + \sqrt{x^2 + a^2}) + C\) | All real numbers |
| \(\displaystyle\frac{1}{\sqrt{x^2 - a^2}}\) | \(\ln|x + \sqrt{x^2 - a^2}| + C\) | \(|x| > a\) |
| \(\displaystyle\frac{1}{\sqrt{a^2 - x^2}}\) | \(\arcsin\displaystyle\frac{x}{a} + C\) | \(|x| < a\) |
| \(\sqrt{a^2 - x^2}\) | \(\displaystyle\frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\arcsin\frac{x}{a} + C\) | \(-a ≤ x ≤ a\) |
| \(\sqrt{x^2 + a^2}\) | \(\displaystyle\frac{x}{2}\sqrt{x^2+a^2} + \frac{a^2}{2}\ln(x + \sqrt{x^2+a^2}) + C\) | All real numbers |
| \(\sqrt{x^2 - a^2}\) | \(\displaystyle\frac{x}{2}\sqrt{x^2-a^2} - \frac{a^2}{2}\ln|x + \sqrt{x^2-a^2}| + C\) | \(|x| ≥ a\) |
| \(\displaystyle\frac{x}{\sqrt{x^2 + a^2}}\) | \(\sqrt{x^2 + a^2} + C\) | All real numbers |
| \(\displaystyle\frac{x}{\sqrt{x^2 - a^2}}\) | \(\sqrt{x^2 - a^2} + C\) | \(|x| > a\) |
| \(\displaystyle\frac{x}{\sqrt{a^2 - x^2}}\) | \(-\sqrt{a^2 - x^2} + C\) | \(|x| < a\) |
| \(\sqrt{ax + b}\) | \(\displaystyle\frac{2(ax + b)^{3/2}}{3a} + C\) | \(ax + b ≥ 0, a ≠ 0\) |
| \(\displaystyle\frac{1}{\sqrt{ax + b}}\) | \(\displaystyle\frac{2\sqrt{ax + b}}{a} + C\) | \(ax + b > 0, a ≠ 0\) |
| \(x\sqrt{x^2 + a^2}\) | \(\displaystyle\frac{(x^2 + a^2)^{3/2}}{3} + C\) | All real numbers |
Products with x
| Function f(x) | Integral ∫f(x)dx | Domain |
|---|---|---|
| \(x\sin x\) | \(\sin x - x\cos x + C\) | All real numbers |
| \(x\cos x\) | \(\cos x + x\sin x + C\) | All real numbers |
| \(x^2\sin x\) | \((2 - x^2)\cos x + 2x\sin x + C\) | All real numbers |
| \(x^2\cos x\) | \((x^2 - 2)\sin x + 2x\cos x + C\) | All real numbers |
| \(x \ln x\) | \(\displaystyle\frac{x^2 \ln x}{2} - \frac{x^2}{4} + C\) | \(x > 0\) |
| \(x^2 \ln x\) | \(\displaystyle\frac{x^3 \ln x}{3} - \frac{x^3}{9} + C\) | \(x > 0\) |
| \(x^n \ln x\) | \(\displaystyle\frac{x^{n+1} \ln x}{n+1} - \frac{x^{n+1}}{(n+1)^2} + C\) | \(x > 0, n ≠ -1\) |
| \(x \arcsin x\) | \(\displaystyle\frac{x^2 \arcsin x}{2} + \frac{\sqrt{1-x^2}}{2} - \frac{x}{2} + C\) | \(-1 ≤ x ≤ 1\) |
| \(x \arctan x\) | \(\displaystyle\frac{x^2 \arctan x}{2} - \frac{x}{2} + \frac{\arctan x}{2} + C\) | All real numbers |
| \(x \sinh x\) | \(x \cosh x - \sinh x + C\) | All real numbers |
| \(x \cosh x\) | \(x \sinh x - \cosh x + C\) | All real numbers |
| \(x^n e^{ax}\) | \(\displaystyle\frac{x^n e^{ax}}{a} - \frac{n}{a}\int x^{n-1} e^{ax} dx\) | All real numbers, \(a ≠ 0\) |
| \(x^3 e^x\) | \((x^3 - 3x^2 + 6x - 6)e^x + C\) | All real numbers |
| \(x\tan x\) | \(x\ln|\cos x| + \displaystyle\frac{x^2}{2} + C\) | \(x ≠ \frac{\pi}{2} + \pi n\) |
Basic Integration Rules
Linearity:
\(\displaystyle\int [af(x) + bg(x)] dx = a\int f(x) dx + b\int g(x) dx\)
\(\displaystyle\int [af(x) + bg(x)] dx = a\int f(x) dx + b\int g(x) dx\)
Integration by parts:
\(\displaystyle\int u \, dv = uv - \int v \, du\)
\(\displaystyle\int u \, dv = uv - \int v \, du\)
Substitution:
\(\displaystyle\int f(\varphi(x))\varphi'(x) dx = \int f(u) du\), where \(u = \varphi(x)\)
\(\displaystyle\int f(\varphi(x))\varphi'(x) dx = \int f(u) du\), where \(u = \varphi(x)\)
Even functions:
\(\displaystyle\int_{-a}^{a} f(x) dx = 2\int_{0}^{a} f(x) dx\) (if \(f(-x) = f(x)\))
\(\displaystyle\int_{-a}^{a} f(x) dx = 2\int_{0}^{a} f(x) dx\) (if \(f(-x) = f(x)\))
Odd functions:
\(\displaystyle\int_{-a}^{a} f(x) dx = 0\) (if \(f(-x) = -f(x)\))
\(\displaystyle\int_{-a}^{a} f(x) dx = 0\) (if \(f(-x) = -f(x)\))
Fundamental Theorem of Calculus:
\(\displaystyle\int_a^b f(x) dx = F(b) - F(a)\), where \(F'(x) = f(x)\)
\(\displaystyle\int_a^b f(x) dx = F(b) - F(a)\), where \(F'(x) = f(x)\)
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Geometric Interpretation of Integration
Comprehensive Mathematical Integration Reference
This integrals table provides a complete reference guide for indefinite integrals of common mathematical functions. The table organizes integral formulas by function type, including basic polynomials, exponential and logarithmic functions, trigonometric functions, inverse trigonometric functions, and advanced expressions involving radicals and rational functions.
General Integration Formula:
∫ f(x)dx = F(x) + C
Where F(x) is the antiderivative of f(x) and C is the constant of integration.
∫ f(x)dx = F(x) + C
Where F(x) is the antiderivative of f(x) and C is the constant of integration.
Each entry in the reference table includes the original function f(x), its corresponding integral ∫f(x)dx, and the domain of validity. The table covers fundamental integration rules, power rule applications, trigonometric identities, logarithmic and exponential integrations, and advanced techniques for radical expressions.
Function Categories Covered
Basic Functions
Constants, powers xn, reciprocals 1/x, square roots √x
Exponential & Logarithmic
ex, ax, ln(x), loga(x), xex
Trigonometric
sin(x), cos(x), tan(x), sec2(x), sin2(x), cos2(x)
Inverse Trigonometric
1/√(1-x2), 1/(1+x2), arcsin(x), arctan(x)
Advanced Expressions
√(a2-x2), 1/√(x2±a2), rational functions
Reference Table Usage Examples
- Looking up ∫x3dx = x4/4 + C for polynomial integration
- Finding ∫e2xdx = e2x/2 + C for exponential functions
- Referencing ∫sin(x)dx = -cos(x) + C for trigonometric integration
- Checking ∫1/√(1-x2)dx = arcsin(x) + C for inverse trig functions
- Locating ∫ln(x)dx = x ln(x) - x + C for logarithmic functions
- Finding ∫1/(x2+4)dx = (1/2)arctan(x/2) + C for rational expressions
- Looking up ∫√(9-x2)dx for radical integration formulas
- Referencing ∫x·cos(x)dx = cos(x) + x·sin(x) + C for products
- Checking ∫sec2(x)dx = tan(x) + C for secant functions
- Finding ∫1/√(x2+1)dx = ln(x + √(x2+1)) + C for hyperbolic forms
The reference includes domain restrictions for each integral, integration rules such as linearity and substitution methods, and visual representation showing integration as the area under a curve. Search functionality allows quick lookup of specific function types or mathematical expressions.