Integral Calculator

Calculate indefinite and definite integrals of polynomial, exponential, and logarithmic functions.

∫axⁿ dx = (a/(n+1))xⁿ⁺^1 + C; ∫eˣ dx = eˣ + C; ∫1/x dx = ln|x| + C; Definite: ∫[a,b]f = F(b) - F(a)
∫x^2 dx = x^3/3 + C; ∫[0,2]x^2 dx = 8/3 - 0 = 2.667; ∫eˣ dx = eˣ + C

What is an integral and what does it represent?

An integral is the "anti-derivative" - reverse of differentiation. It represents accumulation or area under a curve. Indefinite integral ∫f(x)dx = F(x) + C (antiderivative + constant). Definite integral ∫[a,b]f(x)dx = F(b) - F(a) (specific area/accumulation). Example: Velocity -> integral = distance traveled. Acceleration → integral = velocity. Used in: Area calculation, volume, probability, physics.

What are the basic integration rules?

Power rule: ∫xⁿ dx = xⁿ⁺^1/(n+1) + C (n ≠ -1). Constant: ∫k dx = kx + C. Sum: ∫(f+g) dx = ∫f dx + ∫g dx. Examples: ∫x^2 dx = x^3/3 + C, ∫(3x^2 + 2x) dx = x^3 + x^2 + C, ∫5 dx = 5x + C. Special cases: ∫1/x dx = ln|x| + C, ∫eˣ dx = eˣ + C. Always add +C for indefinite integrals.

What is the difference between definite and indefinite integrals?

Indefinite integral ∫f(x)dx: General antiderivative, includes +C, result is a function. Example: ∫x dx = x^2/2 + C. Definite integral ∫[a,b]f(x)dx: Specific numerical value (area), bounds [a,b], no +C needed. Example: ∫[0,2]x dx = [x^2/2] from 0 to 2 = 4/2 - 0 = 2. Definite = evaluate indefinite at bounds and subtract.

When do you use integrals in real life?

Area/Volume: Find area under curves, volume of irregular shapes. Physics: Work (force x distance), center of mass, moment of inertia. Probability: Area under probability density function = probability. Economics: Consumer/producer surplus, total revenue. Engineering: Fluid flow, electric charge. Example: Car accelerates at a(t) = 2t m/s^2. Distance = ∫∫a dt dt = t^3/3 meters.