Integral Calculator
Finding the area under a curve or reversing a derivative by hand can eat up hours of your time. One misplaced exponent or sign error forces you to restart the entire problem. An online integral calculator removes that friction: you input the function, set the bounds if needed, and receive a fully worked solution in seconds.
Quick Examples
The calculator parses standard mathematical notation, recognizing polynomials, trigonometric functions, logarithms, and exponential expressions. In indefinite mode, the engine applies antiderivative rules to return a family of functions. In definite mode, it evaluates the antiderivative at the upper and lower bounds and subtracts to produce a single numeric value. It also handles limits for improper integrals when bounds are infinite.
How the Integral Calculator Processes Functions
When you supply an expression, the calculator translates it into a symbolic tree. It then matches the structure against a library of integration rules: the power rule for polynomials, the chain rule for nested functions, and transformation tables for trigonometric identities.
For definite integrals, the tool computes the antiderivative first, substitutes the upper bound, substitutes the lower bound, and subtracts the second value from the first. If the bounds contain infinity or the integrand has a discontinuity inside the interval, the calculator treats the problem as an improper integral and evaluates the corresponding limit automatically.
Definite vs Indefinite Integrals: Key Distinctions
Not every integral returns a number. Understanding the two main forms keeps your calculations accurate.
| Feature | Indefinite Integral | Definite Integral |
|---|---|---|
| Notation | ∫ f(x) dx | ∫ₐᵇ f(x) dx |
| Result | F(x) + C | A real number |
| Bounds | None | Lower (a) and upper (b) |
| Interpretation | Family of antiderivatives | Net signed area under the curve |
An indefinite integral describes every possible function whose derivative equals the integrand. A definite integral, by contrast, measures the cumulative net area between the curve and the x-axis across a closed interval.
Core Integration Rules You Should Know
Even with automation, knowing why an answer appears helps you catch input mistakes. The calculator relies on the same rules taught in a standard calculus course.
- Power rule. For any real n ≠ -1, ∫ x^n dx = x^(n+1)/(n+1) + C.
- Constant multiple and sum. ∫ [a·f(x) + b·g(x)] dx = a·∫ f(x) dx + b·∫ g(x) dx.
- U-substitution. Reverses the chain rule by replacing an inner function g(x) with a variable u, then adjusting dx accordingly.
- Integration by parts. Derived from the product rule: ∫ u dv = u·v − ∫ v du.
- Partial fractions. Decomposes rational functions into simpler quotients that integrate to natural logarithms.
Worked Examples from Basic to Advanced
Here is how the calculator handles common requests. You can use these expressions to test the tool above.
- Polynomial. ∫ x^2 dx. The power rule gives x^3/3 + C. Evaluated from 0 to 2, the definite result is 8/3 ≈ 2.667.
- Reciprocal. ∫ 1/x dx. The power rule fails here because n = -1 would require division by zero. The antiderivative is ln|x| + C.
- Exponential. ∫ e^x dx. Because the derivative of e^x is itself, the antiderivative is e^x + C.
- Trigonometric, definite. ∫₀^π sin(x) dx. The antiderivative is −cos(x). Evaluating: [−cos(π)] − [−cos(0)] = −(−1) + 1 = 2.
Evaluating Integrals Over Infinite Intervals
An integral becomes improper when one or both bounds are infinite or when the integrand spikes to infinity inside the interval. To evaluate it manually, replace the problematic bound with a variable, compute the definite integral, and take the limit as the variable approaches infinity.
The calculator automates this limit process. For example, ∫₁^∞ 1/x^2 dx is interpreted as the limit of ∫₁^b 1/x^2 dx as b → ∞, which converges to exactly 1. If the limit does not exist, the integral diverges.
What Is the Fastest Way to Verify an Integral?
Differentiation is the built-in error checker. If you integrated correctly, the derivative of your antiderivative must match the original function.
For definite integrals, you can also estimate the area under the curve with a Riemann sum or the trapezoidal rule. Confirm that the numerical approximation aligns with the exact symbolic result. The calculator above shows both the symbolic steps and the final numeric value, making cross-checking immediate.
How Accurate Are Online Integral Results?
Symbolic calculators produce exact answers by manipulating algebraic expressions rather than approximating decimals. They return surds such as √2 or exact logarithms such as ln(3) instead of truncated floats.
When you request a decimal approximation, the accuracy depends on the floating-point precision of the engine, typically 10 to 15 significant digits. For most calculus homework and engineering estimates, this precision is more than sufficient.
Common Mistakes to Avoid
A few syntax errors trip users up. Use explicit multiplication between coefficients and variables: write 2*x instead of 2x. Enclose function arguments in parentheses: sin(x) rather than sin x. For rational functions, parenthesize the denominator: 1/(x+1). Exponentials are entered as e^x or exp(x) depending on the parser. Getting the syntax right ensures the calculator returns the correct antiderivative rather than an error message.