Integral Calculator
Integral calculator: indefinite integrals by the table, by parts and partial fractions; definite integrals by the Newton–Leibniz formula with the area shaded.
The integral calculator finds indefinite integrals with a step-by-step solution and evaluates definite integrals by the Newton–Leibniz formula. It uses the table of integrals, linearity, partial fractions, completing the square and integration by parts. Every antiderivative it finds is checked by differentiation. If the antiderivative is not elementary (for example, ∫e−x²dx), a definite integral is computed numerically.
Table of integrals
| ∫xⁿdx = xⁿ⁺¹/(n + 1) + C | ∫dx/x = ln|x| + C |
| ∫eˣdx = eˣ + C | ∫aˣdx = aˣ/ln a + C |
| ∫sin x dx = −cos x + C | ∫cos x dx = sin x + C |
| ∫dx/cos²x = tan x + C | ∫dx/sin²x = −cot x + C |
| ∫dx/(x² + a²) = (1/a) arctan(x/a) + C | ∫dx/√(a² − x²) = arcsin(x/a) + C |
For a linear argument: ∫f(ax + b)dx = F(ax + b)/a + C. For example, ∫sin 2x dx = −cos 2x / 2 + C.
Integration by parts
∫u dv = uv − ∫v du
Example. ∫x eˣ dx: u = x, dv = eˣdx ⇒ x eˣ − ∫eˣdx = x eˣ − eˣ + C.
Definite integral
∫ab f(x)dx = F(b) − F(a)
Example. ∫0π sin x dx = −cos π + cos 0 = 2.
For a non-negative function the definite integral is the area under the graph — the calculator shades it on the plot.
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