Integral Calculator

Introduction to Integrals

Integration is the inverse operation of differentiation, used to calculate the area under curves, volume of objects, and many other applications in mathematics and physics.

Indefinite Integral

Enter function f(x):

∫ (x² + 3x + 1) dx

Definite Integral

Enter function f(x):

Lower bound (a):

Upper bound (b):

∫₀² x² dx

Basic Integration Formulas

Power Function

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C

(n ≠ -1)

Function 1/x

∫ 1/x dx = ln|x| + C

Exponential Function

∫ eˣ dx = eˣ + C

∫ aˣ dx = aˣ/ln(a) + C

Sine Function

∫ sin(x) dx = -cos(x) + C

Cosine Function

∫ cos(x) dx = sin(x) + C

Tangent Function

∫ tan(x) dx = -ln|cos(x)| + C

Secant² Function

∫ sec²(x) dx = tan(x) + C

Cosecant² Function

∫ csc²(x) dx = -cot(x) + C

Substitution Method

∫ f(g(x))·g'(x) dx

= ∫ f(u) du (u = g(x))

Integration by Parts

∫ u·dv = u·v - ∫ v·du