Integrals Table
Complete reference table of basic indefinite integrals organized by category: polynomial, exponential, trigonometric, and inverse trigonometric.
C is an arbitrary constant of integration
Polynomial & Power
| Integrand | Integral |
|---|---|
| ∫ a dx | ax + C |
| ∫ x dx | x²/2 + C |
| ∫ xⁿ dx (n ≠ −1) | xⁿ⁺¹/(n+1) + C |
| ∫ 1/x dx | ln|x| + C |
| ∫ 1/√x dx | 2√x + C |
Exponential & Logarithmic
| Integrand | Integral |
|---|---|
| ∫ eˣ dx | eˣ + C |
| ∫ aˣ dx | aˣ / ln(a) + C |
| ∫ ln(x) dx | x·ln(x) − x + C |
Trigonometric
| Integrand | Integral |
|---|---|
| ∫ sin(x) dx | −cos(x) + C |
| ∫ cos(x) dx | sin(x) + C |
| ∫ tan(x) dx | −ln|cos(x)| + C |
| ∫ cot(x) dx | ln|sin(x)| + C |
| ∫ 1/cos²(x) dx | tan(x) + C |
| ∫ 1/sin²(x) dx | −cot(x) + C |
| ∫ sin²(x) dx | x/2 − sin(2x)/4 + C |
| ∫ cos²(x) dx | x/2 + sin(2x)/4 + C |
Inverse Trigonometric
| Integrand | Integral |
|---|---|
| ∫ 1/(1 + x²) dx | arctan(x) + C |
| ∫ 1/√(1 − x²) dx | arcsin(x) + C |
| ∫ −1/√(1 − x²) dx | arccos(x) + C |
| ∫ 1/(x² + a²) dx | (1/a)·arctan(x/a) + C |
| ∫ 1/√(a² − x²) dx | arcsin(x/a) + C |
How to Use the Integrals Table
- Browse the reference table to find the values you need.
- Use search or scroll to locate specific entries.
- Click on a value to copy it or see more details.
- Use the table as a quick reference during calculations or study.
Schnellreferenz
| Von | Nach |
|---|---|
| 1 × 1 | 1 |
| 5 × 5 | 25 |
| 7 × 8 | 56 |
| 9 × 9 | 81 |
| 12 × 12 | 144 |
| 15 × 15 | 225 |
Anwendungsfälle
- •Quick lookup of values during math class or professional work.
- •Verifying calculations without needing a full scientific calculator.
- •Studying mathematical relationships, patterns, and properties.
- •Using as a handy reference during engineering or science tasks.
Formel
An indefinite integral (antiderivative) F(x) of f(x) satisfies F′(x) = f(x). The general antiderivative is F(x) + C, where C is an arbitrary constant.
Häufig gestellte Fragen
What is an indefinite integral?
An indefinite integral ∫f(x)dx is a function F(x) + C whose derivative is f(x). The constant C accounts for all possible antiderivatives.
Why is there always a +C in indefinite integrals?
The constant C represents all possible vertical shifts of the antiderivative. Since the derivative of a constant is zero, any F(x) + C has the same derivative f(x).
What is the integral of 1/x?
∫(1/x)dx = ln|x| + C. The absolute value is needed because ln(x) is only defined for positive x, but 1/x exists for all x ≠ 0.
How are integrals and derivatives related?
They are inverse operations. The Fundamental Theorem of Calculus states that if F′(x) = f(x), then ∫f(x)dx = F(x) + C. Differentiating an integral returns the original function.