Antiderivatives Table
Complete reference guide for indefinite integrals
Basic Antiderivatives
| Function \(f(x)\) | Antiderivative \(F(x)\) | Domain |
|---|---|---|
| \(k\) (constant) | \(kx + C\) | \(\mathbb{R}\) |
| \(x^n\) (where \(n \ne -1\)) | \(\displaystyle\frac{x^{n+1}}{n+1} + C\) | \(\mathbb{R}\) (if \(n \geq 0\)), \(\mathbb{R}, x \neq 0\) (if \(n < 0\)) |
| \(\displaystyle\frac{1}{x}\) | \(\ln|x| + C\) | \(\mathbb{R}, x \neq 0\) |
| \(\sqrt{x}\) | \(\displaystyle\frac{2\sqrt{x^3}}{3} + C\) | \(x \geq 0\) |
| \(\displaystyle\frac{1}{\sqrt{x}}\) | \(2\sqrt{x} + C\) | \(x > 0\) |
| \(e^x\) | \(e^x + C\) | \(\mathbb{R}\) |
| \(a^x\) (where \(a > 0, a \ne 1\)) | \(\displaystyle\frac{a^x}{\ln a} + C\) | \(\mathbb{R}\) |
| \(\ln x\) | \(x \ln x - x + C\) | \(x > 0\) |
| \(\log_a x\) | \(\displaystyle\frac{x \ln x - x}{\ln a} + C\) | \(x > 0, a > 0, a \ne 1\) |
Function f(x)
\(k\) (constant)
Antiderivative F(x)
\(kx + C\)
Domain
\(\mathbb{R}\)
Function f(x)
\(x^n\) (where \(n \ne -1\))
Antiderivative F(x)
\(\displaystyle\frac{x^{n+1}}{n+1} + C\)
Domain
\(\mathbb{R}\) (if \(n \geq 0\)), \(\mathbb{R}, x \ne 0\) (if \(n < 0\))
Function f(x)
\(\displaystyle\frac{1}{x}\)
Antiderivative F(x)
\(\ln|x| + C\)
Domain
\(\mathbb{R}, x \ne 0\)
Function f(x)
\(\sqrt{x}\)
Antiderivative F(x)
\(\displaystyle\frac{2\sqrt{x^3}}{3} + C\)
Domain
\(x \geq 0\)
Function f(x)
\(\displaystyle\frac{1}{\sqrt{x}}\)
Antiderivative F(x)
\(2\sqrt{x} + C\)
Domain
\(x > 0\)
Function f(x)
\(e^x\)
Antiderivative F(x)
\(e^x + C\)
Domain
\(\mathbb{R}\)
Function f(x)
\(a^x\) (where \(a > 0, a \ne 1\))
Antiderivative F(x)
\(\displaystyle\frac{a^x}{\ln a} + C\)
Domain
\(\mathbb{R}\)
Function f(x)
\(\ln x\)
Antiderivative F(x)
\(x \ln x - x + C\)
Domain
\(x > 0\)
Function f(x)
\(\log_a x\)
Antiderivative F(x)
\(\displaystyle\frac{x \ln x - x}{\ln a} + C\)
Domain
\(x > 0, a > 0, a \ne 1\)
Trigonometric Functions
| Function \(f(x)\) | Antiderivative \(F(x)\) | Domain |
|---|---|---|
| \(\sin x\) | \(-\cos x + C\) | \(\mathbb{R}\) |
| \(\cos x\) | \(\sin x + C\) | \(\mathbb{R}\) |
| \(\tan x\) | \(-\ln|\cos x| + C\) | \(x \ne \frac{\pi}{2} + \pi n\) |
| \(\cot x\) | \(\ln|\sin x| + C\) | \(x \ne \pi n\) |
| \(\displaystyle\frac{1}{\cos^2 x} = \sec^2 x\) | \(\tan x + C\) | \(x \ne \frac{\pi}{2} + \pi n\) |
| \(\displaystyle\frac{1}{\sin^2 x} = \csc^2 x\) | \(-\cot x + C\) | \(x \ne \pi n\) |
| \(\sin^2 x\) | \(\displaystyle\frac{x}{2} - \frac{\sin 2x}{4} + C\) | \(\mathbb{R}\) |
| \(\cos^2 x\) | \(\displaystyle\frac{x}{2} + \frac{\sin 2x}{4} + C\) | \(\mathbb{R}\) |
| \(\sin x \cos x\) | \(\displaystyle\frac{\sin^2 x}{2} + C = -\frac{\cos^2 x}{2} + C\) | \(\mathbb{R}\) |
Function f(x)
\(\sin x\)
Antiderivative F(x)
\(-\cos x + C\)
Domain
\(\mathbb{R}\)
Function f(x)
\(\cos x\)
Antiderivative F(x)
\(\sin x + C\)
Domain
\(\mathbb{R}\)
Function f(x)
\(\tan x\)
Antiderivative F(x)
\(-\ln|\cos x| + C\)
Domain
\(x \ne \frac{\pi}{2} + \pi n\)
Function f(x)
\(\cot x\)
Antiderivative F(x)
\(\ln|\sin x| + C\)
Domain
\(x \ne \pi n\)
Function f(x)
\(\displaystyle\frac{1}{\cos^2 x} = \sec^2 x\)
Antiderivative F(x)
\(\tan x + C\)
Domain
\(x \ne \frac{\pi}{2} + \pi n\)
Function f(x)
\(\displaystyle\frac{1}{\sin^2 x} = \csc^2 x\)
Antiderivative F(x)
\(-\cot x + C\)
Domain
\(x \ne \pi n\)
Function f(x)
\(\sin^2 x\)
Antiderivative F(x)
\(\displaystyle\frac{x}{2} - \frac{\sin 2x}{4} + C\)
Domain
\(\mathbb{R}\)
Function f(x)
\(\cos^2 x\)
Antiderivative F(x)
\(\displaystyle\frac{x}{2} + \frac{\sin 2x}{4} + C\)
Domain
\(\mathbb{R}\)
Function f(x)
\(\sin x \cos x\)
Antiderivative F(x)
\(\displaystyle\frac{\sin^2 x}{2} + C\)
Domain
\(\mathbb{R}\)
Inverse Trigonometric Functions
| Function \(f(x)\) | Antiderivative \(F(x)\) | Domain |
|---|---|---|
| \(\displaystyle\frac{1}{\sqrt{1-x^2}}\) | \(\arcsin x + C\) | \(-1 < x < 1\) |
| \(\displaystyle-\frac{1}{\sqrt{1-x^2}}\) | \(\arccos x + C\) | \(-1 < x < 1\) |
| \(\displaystyle\frac{1}{1+x^2}\) | \(\arctan x + C\) | \(\mathbb{R}\) |
| \(\displaystyle-\frac{1}{1+x^2}\) | \(\text{arccot } x + C\) | \(\mathbb{R}\) |
| \(\arcsin x\) | \(x \arcsin x + \sqrt{1-x^2} + C\) | \(-1 \leq x \leq 1\) |
| \(\arccos x\) | \(x \arccos x - \sqrt{1-x^2} + C\) | \(-1 \leq x \leq 1\) |
| \(\arctan x\) | \(x \arctan x - \displaystyle\frac{1}{2}\ln(1+x^2) + C\) | \(\mathbb{R}\) |
Function f(x)
\(\displaystyle\frac{1}{\sqrt{1-x^2}}\)
Antiderivative F(x)
\(\arcsin x + C\)
Domain
\(-1 < x < 1\)
Function f(x)
\(\displaystyle-\frac{1}{\sqrt{1-x^2}}\)
Antiderivative F(x)
\(\arccos x + C\)
Domain
\(-1 < x < 1\)
Function f(x)
\(\displaystyle\frac{1}{1+x^2}\)
Antiderivative F(x)
\(\arctan x + C\)
Domain
\(\mathbb{R}\)
Function f(x)
\(\displaystyle-\frac{1}{1+x^2}\)
Antiderivative F(x)
\(\text{arccot } x + C\)
Domain
\(\mathbb{R}\)
Function f(x)
\(\arcsin x\)
Antiderivative F(x)
\(x \arcsin x + \sqrt{1-x^2} + C\)
Domain
\(-1 \leq x \leq 1\)
Function f(x)
\(\arccos x\)
Antiderivative F(x)
\(x \arccos x - \sqrt{1-x^2} + C\)
Domain
\(-1 \leq x \leq 1\)
Function f(x)
\(\arctan x\)
Antiderivative F(x)
\(x \arctan x - \displaystyle\frac{1}{2}\ln(1+x^2) + C\)
Domain
\(\mathbb{R}\)
Useful Antiderivatives
| Function \(f(x)\) | Antiderivative \(F(x)\) | Domain |
|---|---|---|
| \(\displaystyle\frac{1}{\sqrt{x^2 + a^2}}\) | \(\ln(x + \sqrt{x^2 + a^2}) + C\) | \(\mathbb{R}\) |
| \(\displaystyle\frac{1}{\sqrt{x^2 - a^2}}\) | \(\ln|x + \sqrt{x^2 - a^2}| + C\) | \(|x| > a\) |
| \(\displaystyle\frac{1}{\sqrt{a^2 - x^2}}\) | \(\arcsin\displaystyle\frac{x}{a} + C\) | \(|x| < a\) |
| \(\displaystyle\frac{1}{x^2 + a^2}\) | \(\displaystyle\frac{1}{a} \arctan\frac{x}{a} + C\) | \(\mathbb{R}\) |
| \(\displaystyle\frac{1}{x^2 - a^2}\) | \(\displaystyle\frac{1}{2a} \ln\left|\frac{x-a}{x+a}\right| + C\) | \(x \ne +a, x \ne -a\) |
| \(\displaystyle\frac{1}{a^2 - x^2}\) | \(\displaystyle\frac{1}{2a} \ln\left|\frac{a+x}{a-x}\right| + C\) | \(x \ne +a, x \ne -a\) |
| \(\sqrt{a^2 - x^2}\) | \(\displaystyle\frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\arcsin\frac{x}{a} + C\) | \(-a \leq x \leq a\) |
| \(\sqrt{x^2 + a^2}\) | \(\displaystyle\frac{x}{2}\sqrt{x^2+a^2} + \frac{a^2}{2}\ln(x + \sqrt{x^2+a^2}) + C\) | \(\mathbb{R}\) |
| \(\sqrt{x^2 - a^2}\) | \(\displaystyle\frac{x}{2}\sqrt{x^2-a^2} - \frac{a^2}{2}\ln|x + \sqrt{x^2-a^2}| + C\) | \(|x| \geq a\) |
| \(xe^x\) | \((x-1)e^x + C\) | \(\mathbb{R}\) |
| \(x\sin x\) | \(\sin x - x\cos x + C\) | \(\mathbb{R}\) |
| \(x\cos x\) | \(\cos x + x\sin x + C\) | \(\mathbb{R}\) |
Function f(x)
\(\displaystyle\frac{1}{\sqrt{x^2 + a^2}}\)
Antiderivative F(x)
\(\ln(x + \sqrt{x^2 + a^2}) + C\)
Domain
\(\mathbb{R}\)
Function f(x)
\(\displaystyle\frac{1}{\sqrt{x^2 - a^2}}\)
Antiderivative F(x)
\(\ln|x + \sqrt{x^2 - a^2}| + C\)
Domain
\(|x| > a\)
Function f(x)
\(\displaystyle\frac{1}{\sqrt{a^2 - x^2}}\)
Antiderivative F(x)
\(\arcsin\displaystyle\frac{x}{a} + C\)
Domain
\(|x| < a\)
Function f(x)
\(\displaystyle\frac{1}{x^2 + a^2}\)
Antiderivative F(x)
\(\displaystyle\frac{1}{a} \arctan\frac{x}{a} + C\)
Domain
\(\mathbb{R}\)
Function f(x)
\(\displaystyle\frac{1}{x^2 - a^2}\)
Antiderivative F(x)
\(\displaystyle\frac{1}{2a} \ln\left|\frac{x-a}{x+a}\right| + C\)
Domain
\(x \ne +a, x \ne -a\)
Function f(x)
\(\displaystyle\frac{1}{a^2 - x^2}\)
Antiderivative F(x)
\(\displaystyle\frac{1}{2a} \ln\left|\frac{a+x}{a-x}\right| + C\)
Domain
\(x \ne +a, x \ne -a\)
Function f(x)
\(xe^x\)
Antiderivative F(x)
\((x-1)e^x + C\)
Domain
\(\mathbb{R}\)
Function f(x)
\(x\sin x\)
Antiderivative F(x)
\(\sin x - x\cos x + C\)
Domain
\(\mathbb{R}\)
Function f(x)
\(x\cos x\)
Antiderivative F(x)
\(\cos x + x\sin x + C\)
Domain
\(\mathbb{R}\)
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 \(f(-x) = -f(x)\))
Important Notes
• \(C\) is an arbitrary constant of integration
• All antiderivatives are determined up to a constant \(C\)
• When evaluating definite integrals, the constant \(C\) cancels out
• Domain refers to the domain of the integrand function
• \(\mathbb{R}\) denotes the set of all real numbers
Useful Techniques
Trigonometric substitutions:
• For \(\sqrt{a^2 - x^2}\): \(x = a\sin t\)
• For \(\sqrt{x^2 + a^2}\): \(x = a\tan t\)
• For \(\sqrt{x^2 - a^2}\): \(x = a\sec t\)
• For \(\sqrt{a^2 - x^2}\): \(x = a\sin t\)
• For \(\sqrt{x^2 + a^2}\): \(x = a\tan t\)
• For \(\sqrt{x^2 - a^2}\): \(x = a\sec t\)
Rational function integration:
Decompose into partial fractions and integrate each separately
Decompose into partial fractions and integrate each separately
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 \(F'(x) = f(x)\)
Weierstrass substitution:
\(t = \tan\frac{x}{2}\), then \(\sin x = \frac{2t}{1+t^2}\), \(\cos x = \frac{1-t^2}{1+t^2}\)
\(t = \tan\frac{x}{2}\), then \(\sin x = \frac{2t}{1+t^2}\), \(\cos x = \frac{1-t^2}{1+t^2}\)
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Antiderivatives Table - Complete Reference Guide
This comprehensive antiderivatives table serves as a complete reference guide for indefinite integrals, featuring an advanced search system to quickly locate specific mathematical functions and their antiderivatives. The table covers fundamental calculus operations where you can find the antiderivative F(x) for any given function f(x).
Mathematical Foundation
An antiderivative of a function f(x) is a function F(x) whose derivative equals the original function. The relationship is expressed as:
F'(x) = f(x), therefore ∫f(x)dx = F(x) + C
where C represents the constant of integration.
Table Features
• Smart Search System: Type keywords like "sin", "logarithm", "square root" to instantly filter results
• Mobile-Responsive Design: Optimized viewing on all devices with adaptive card layouts
• Domain Information: Each entry includes the valid domain for the function
• Mathematical Notation: Professional LaTeX rendering for precise mathematical display
• Categorized Sections: Organized by function types for systematic reference
Function Categories Covered
The table includes four main categories of functions:
Basic Functions: Power functions (xn), exponential functions (ex, ax), logarithmic functions (ln x, loga x), and square root functions (√x, 1/√x).
Trigonometric Functions: Standard trigonometric functions including sin x, cos x, tan x, cot x, and their squares (sin²x, cos²x).
Inverse Trigonometric Functions: Antiderivatives involving arcsin, arccos, arctan, and related expressions like 1/√(1-x²) and 1/(1+x²).
Advanced Functions: Complex expressions involving radicals (√(x²±a²)), rational functions (1/(x²±a²)), and products like xex, x sin x.
Usage Examples
1. Find ∫x³ dx → Result: x⁴/4 + C
2. Find ∫sin x dx → Result: -cos x + C
3. Find ∫1/x dx → Result: ln|x| + C
4. Find ∫e^x dx → Result: e^x + C
5. Find ∫1/√(1-x²) dx → Result: arcsin x + C
6. Find ∫1/(1+x²) dx → Result: arctan x + C
7. Find ∫√x dx → Result: (2√x³)/3 + C
8. Find ∫cos²x dx → Result: x/2 + sin(2x)/4 + C
9. Find ∫ln x dx → Result: x ln x - x + C
10. Find ∫tan x dx → Result: -ln|cos x| + C
Search Functionality
The integrated search system recognizes multiple input formats and synonyms. You can search using mathematical terms (sine, cosine, logarithm), abbreviated forms (sin, cos, ln), or descriptive terms (square root, exponential). The search instantly highlights matching entries across all categories, making it efficient to locate specific antiderivatives.
Integration Rules Reference
The table also includes essential integration rules such as linearity property, integration by parts formula, substitution method, and special properties for even and odd functions. These rules complement the antiderivative formulas and provide a complete integration reference.