liminfoIntegral Reference Table
Free web tool: Integral Reference Table
About Integral Reference Table
The Integral Reference Table is a free, searchable calculus reference covering 37 standard integral formulas organized across six categories: Basic (power rule, constant, square root), Exponential (eˣ, aˣ, x·eˣ), Logarithmic (ln x, log_a x), Trigonometric (sin, cos, tan, cot, sec, csc, and their squares), Inverse Trigonometric (arcsin, arccos, arctan forms), and Hyperbolic (sinh, cosh, tanh, sech, and inverse hyperbolic). Each entry displays the integrand f(x), the integral symbol, and the antiderivative result with the constant of integration C.
Calculus students, engineering undergraduates, physicists, and teachers use this table to quickly look up antiderivatives while solving differential equations, computing areas and volumes, or reviewing for exams. The category filter buttons let you focus on one class of functions at a time, while the search box matches against function notation so you can type "sin" to see all sine-family integrals or "ln" to find logarithmic forms instantly.
The tool renders all 37 formulas entirely in the browser using Unicode mathematical notation — no LaTeX engine or MathJax is required. Formula entries use superscript (ⁿ, ²) and subscript characters to approximate mathematical typesetting in plain text. Filtering is performed with React useMemo so results update instantly without any debounce delay. No account, no download, and no server interaction is needed.
Key Features
- 37 standard integral formulas covering all essential calculus categories
- Category filter tabs: Basic, Exponential, Logarithmic, Trigonometric, Inverse Trig, Hyperbolic
- Real-time search by function notation — type "cos", "arctan", or "sinh" to filter instantly
- Formula count display showing how many results match the current filter
- Integrand and antiderivative displayed in monospace font for clear mathematical reading
- Integral symbol (∫) and dx notation rendered inline with each formula entry
- Category badge on each formula card for visual classification at a glance
- Fully client-side with Unicode math characters — no LaTeX or MathJax needed
Frequently Asked Questions
What is the power rule for integration?
The power rule states that ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C for any real n except n = -1. When n = -1, the integral is ∫ 1/x dx = ln|x| + C. This is the most fundamental integration formula and is the basis for integrating polynomial functions.
How do you integrate eˣ and other exponential functions?
The integral of eˣ is eˣ + C — the exponential function is its own antiderivative. For aˣ (where a > 0 and a ≠ 1), the integral is aˣ/ln(a) + C. The integral of x·eˣ requires integration by parts and equals (x-1)eˣ + C.
What is the integral of sin(x) and cos(x)?
∫ sin(x) dx = -cos(x) + C and ∫ cos(x) dx = sin(x) + C. Note the sign change for sine. The integral of tan(x) is -ln|cos(x)| + C, and ∫ cot(x) dx = ln|sin(x)| + C. For sec²(x) and csc²(x), the results are tan(x) + C and -cot(x) + C respectively.
What are the inverse trigonometric integrals?
The key inverse trig integrals are: ∫ 1/√(1-x²) dx = arcsin(x) + C; ∫ -1/√(1-x²) dx = arccos(x) + C; ∫ 1/(1+x²) dx = arctan(x) + C; and the generalized form ∫ 1/(a²+x²) dx = (1/a)arctan(x/a) + C. These arise frequently in problems involving circles and trigonometric substitutions.
What is the difference between hyperbolic and trigonometric integrals?
Hyperbolic functions (sinh, cosh, tanh) have integrals structurally similar to trig but without sign alternation: ∫ sinh(x) dx = cosh(x) + C and ∫ cosh(x) dx = sinh(x) + C (both positive). By contrast, ∫ sin(x) dx = -cos(x) + C (negative). The inverse hyperbolic integrals ∫ 1/√(x²+1) dx = arcsinh(x) + C resemble inverse trig forms.
How do I integrate ln(x)?
The integral of ln(x) is x·ln(x) - x + C. This is derived using integration by parts with u = ln(x) and dv = dx. Similarly, ∫ log_a(x) dx = x·log_a(x) - x/ln(a) + C, which reduces to the natural log form when a = e.
How do I integrate sin²(x) and cos²(x)?
Use the half-angle identities: ∫ sin²(x) dx = x/2 - sin(2x)/4 + C and ∫ cos²(x) dx = x/2 + sin(2x)/4 + C. These are derived from the identities sin²(x) = (1 - cos(2x))/2 and cos²(x) = (1 + cos(2x))/2, which convert the squared functions into integrable forms.
What does the constant of integration C mean?
The constant C in every indefinite integral represents the fact that antiderivatives are not unique — any constant can be added to an antiderivative and the derivative will still equal the original function. When you compute a definite integral over an interval [a, b], the constants cancel out. For initial value problems, you solve for C using the given condition.