Integral Calculator
The tool below computes an integral — either an indefinite integral (antiderivative + C) or a definite integral evaluated between two bounds. Enter the integrand and the calculator picks the right technique and shows the work.
Type naturally or click ⌨ in the box to use the math keyboard.
For an indefinite integral use \int f(x)\,dx; for a definite integral add bounds: \int_{a}^{b} f(x)\,dx.
An integral computes the accumulated effect of a function — either as an antiderivative (indefinite) or as the signed area under a curve between two bounds (definite). For example, \(\int x^2 , dx = \dfrac{x^3}{3} + C\), and \(\int_0^1 x^2 , dx = \dfrac{1}{3}\).
Worked Examples
Three examples worked by hand — the same logic the calculator uses, just spelled out so you can follow it on paper.
Example 1: Power rule — evaluate \(\int x^4 , dx\)
The power rule for integration is \(\int x^n , dx = \dfrac{x^{n+1}}{n+1} + C\) for \(n \neq -1\). Add 1 to the exponent, divide by the new exponent, then add the constant:
\[\int x^4 , dx = \dfrac{x^5}{5} + C\]
Check by differentiating: \(\dfrac{d}{dx}\left(\dfrac{x^5}{5}\right) = \dfrac{5x^4}{5} = x^4\). ✓
Example 2: Substitution — evaluate \(\int 2x \cos(x^2) , dx\)
The integrand has a function \(\cos(\cdot)\) wrapped around \(x^2\), and the \(2x\) out front is exactly the derivative of \(x^2\). That's the substitution signal.
Let \(u = x^2\), so \(du = 2x , dx\). Substitute:
\[\int 2x \cos(x^2) , dx = \int \cos(u) , du = \sin(u) + C = \sin(x^2) + C\]
Example 3: Definite integral — evaluate \(\int_0^2 (3x^2 + 1) , dx\)
Step 1: Find the antiderivative.
\[\int (3x^2 + 1) , dx = x^3 + x + C\]
Step 2: Evaluate at the upper bound and subtract the value at the lower bound (the Fundamental Theorem of Calculus):
\[\bigl[x^3 + x\bigr]_0^2 = (8 + 2) - (0 + 0) = 10\]
Definite integrals produce a number, not a function. The "+ C" cancels out in the subtraction.
Indefinite vs. Definite Integrals
An indefinite integral asks for an antiderivative — a function whose derivative is the integrand. Because the derivative of a constant is zero, indefinite integrals always include "+ C," the constant of integration:
\[\int x^2 , dx = \frac{x^3}{3} + C\]
A definite integral evaluates the antiderivative at two specific bounds and subtracts. The result is a number, not a function, and there's no "+ C":
\[\int_{0}^{1} x^2 , dx = \left[\frac{x^3}{3}\right]_0^1 = \frac{1}{3} - 0 = \frac{1}{3}\]
If you only want an antiderivative (no bounds), the Antiderivative Calculator is dedicated to that case.
Try These Examples
\int x^2 \, dx— power rule\int \sin(x) \, dx— basic trig\int e^x \, dx— exponential\int \frac{1}{x} \, dx— natural log\int_{0}^{1} x^2 \, dx— definite integral\int 2x \cdot e^{x^2} \, dx— substitution\int x \cdot e^x \, dx— integration by parts
Common Integration Techniques
A quick reference for the techniques this calculator uses:
| Method | When to use |
|---|---|
| Power rule | \(\int x^n , dx = \dfrac{x^{n+1}}{n+1} + C\) (for \(n \neq -1\)) |
| \(u\)-substitution | A piece of the integrand is the derivative of another piece |
| Integration by parts | \(\int u , dv = uv - \int v , du\) — for products like \(x e^x\) |
| Partial fractions | Rational functions where the denominator factors |
| Trig substitution | Expressions like \(\sqrt{a^2 - x^2}\), \(\sqrt{a^2 + x^2}\), \(\sqrt{x^2 - a^2}\) |
| Trig identities | Powers of sine and cosine, products of trig functions |
For definite integrals, after finding the antiderivative, evaluate at the upper bound and subtract the value at the lower bound — the Fundamental Theorem of Calculus.
Tips for Using the Calculator
- For indefinite integrals, type
\int f(x) \, dx— the\,adds a small space beforedx - For definite integrals, add bounds:
\int_{0}^{1} f(x) \, dx - For powers, use
^:x^3means \(x^3\) - For fractions, use
\frac{a}{b}or justa/b - The virtual math keyboard has an ∫ button for one-click integral entry
Frequently Asked Questions
What does an integral represent geometrically?
A definite integral \(\int_a^b f(x) , dx\) is the signed area between the curve \(y = f(x)\) and the x-axis, from \(x = a\) to \(x = b\). "Signed" means area below the x-axis counts as negative. An indefinite integral has no geometric area attached — it's a family of antiderivatives.
Why is there always a "+ C" in indefinite integrals?
The derivative of any constant is zero, so adding a constant to an antiderivative doesn't change its derivative. Every function has infinitely many antiderivatives differing by a constant. The "+ C" captures all of them at once.
When do I use substitution vs. integration by parts?
Use substitution when one part of the integrand is the derivative of another part — like \(2x\) and \(x^2\). Use integration by parts when the integrand is a product of two functions that aren't related by differentiation, like \(x \cdot e^x\) or \(x \cdot \sin(x)\).
Can every function be integrated?
Every continuous function has an antiderivative, but not every antiderivative can be written using elementary functions (polynomials, exponentials, trig, etc.). For example, \(\int e^{-x^2} , dx\) has no closed-form elementary expression — that's why it's defined as a special function (the error function).
What's the Fundamental Theorem of Calculus?
It links differentiation and integration: if \(F(x)\) is an antiderivative of \(f(x)\), then \(\int_a^b f(x) , dx = F(b) - F(a)\). That's the rule we use to evaluate definite integrals — find an antiderivative, plug in the bounds, subtract.