Integration Solver
Antiderivative Calculator & Integral Solver
An antiderivative (also called an indefinite integral) is the reverse of differentiation. If F'(x) = f(x), then F(x) is an antiderivative of f(x). This solver doubles as an antiderivative calculator and definite integral evaluator — enter ∫ f(x) dx or ∫ₐᵇ f(x) dx and you get the full step-by-step solution.
How to use the antiderivative calculator
- Type
\int x^2 dxfor the antiderivative of x² - Type
\int_0^1 x^2 dxfor the definite integral from 0 to 1 - Type
\int e^x dxfor ∫eˣ dx - Type
\int \sin(x) \cos(x) dxfor substitution-based problems
The most common antiderivatives (every student should memorise)
$\int x^n \, dx = \dfrac{x^{n+1}}{n+1} + C$ (n ≠ −1)
$\int \dfrac{1}{x} \, dx = \ln|x| + C$
$\int e^x \, dx = e^x + C$
$\int \sin(x) \, dx = -\cos(x) + C$
$\int \cos(x) \, dx = \sin(x) + C$
$\int \sec^2(x) \, dx = \tan(x) + C$
Worked example — integration by substitution
Find ∫ 2x cos(x²) dx.
- Step 1. Let u = x². Then du = 2x dx.
- Step 2. The integral becomes ∫ cos(u) du.
- Step 3. Integrate: ∫ cos(u) du = sin(u) + C.
- Step 4. Substitute back: sin(x²) + C.
Worked example — integration by parts
Find ∫ x eˣ dx using the formula ∫ u dv = uv − ∫ v du.
- Choose u and dv (LIATE rule). u = x (algebraic, easier to differentiate), dv = eˣ dx.
- Compute du and v. du = dx, v = eˣ.
- Apply the formula. ∫ x eˣ dx = x eˣ − ∫ eˣ dx = x eˣ − eˣ + C = eˣ(x − 1) + C.
Definite integrals — the Fundamental Theorem of Calculus
If F(x) is an antiderivative of f(x), then ∫ₐᵇ f(x) dx = F(b) − F(a). So ∫₀² x² dx = [x³/3]₀² = 8/3 − 0 = 8/3 ≈ 2.667. Definite integrals give a numerical value (the signed area under the curve); indefinite integrals give a family of functions (the + C captures all the antiderivatives).
Why is there always a + C?
Because differentiation kills constants — the derivative of any constant is zero. So when you reverse the process, you have to add back an unknown constant of integration. The C represents the entire family of curves that all have the same slope at every x.