Integral Calculator

How Integration Works

Integration is the reverse of differentiation — finding a function F(x) such that F'(x) = f(x). The indefinite integral ∫f(x)dx = F(x)+C gives a family of antiderivatives (+C because any constant added to F has the same derivative). The definite integral ∫ₐᵇ f(x)dx = F(b)−F(a) gives the exact signed area under the curve from x=a to x=b (Fundamental Theorem of Calculus Part 2).

Integration Techniques

Direct integration: Match the integrand to a known formula. ∫x⁵dx = x⁶/6+C. ∫cos x dx = sin x+C. ∫eˣdx = eˣ+C. ∫1/x dx = ln|x|+C.

U-Substitution: When the integrand contains a composite function and its inner derivative. ∫2x(x²+1)³dx — let u=x²+1, du=2xdx → ∫u³du = u⁴/4+C = (x²+1)⁴/4+C.

Integration by Parts: ∫u dv = uv − ∫v du. For products of unrelated functions. ∫x·eˣdx → u=x, dv=eˣdx → xeˣ−eˣ+C = eˣ(x−1)+C.

Common Integral Formulas

∫xⁿdx = xⁿ⁺¹/(n+1)+C (n≠−1) · ∫eˣdx = eˣ+C · ∫ln x dx = x·ln x − x+C · ∫sin x dx = −cos x+C · ∫cos x dx = sin x+C · ∫sec²x dx = tan x+C · ∫1/(1+x²)dx = arctan x+C · ∫1/√(1−x²)dx = arcsin x+C

The Fundamental Theorem of Calculus — Why It's Fundamental

The Fundamental Theorem has two parts, both astonishing. Part 1 says that if you define a function by integration — F(x) = ∫ₐˣ f(t)dt — then differentiating it gives back the original function: F'(x) = f(x). Integration and differentiation are inverse operations. Part 2 says the definite integral ∫ₐᵇ f(x)dx = F(b) − F(a) for any antiderivative F. This converts an infinite sum (the Riemann integral) into a simple subtraction. Without this theorem, computing areas would require actually taking limits of Riemann sums every time — impossible for all but the simplest functions.

The deeper insight: the FTC connects the local (derivative = instantaneous rate) to the global (integral = total accumulation). The total change F(b)−F(a) equals the integral of the rate of change F'(x) = f(x). This relationship — total change = accumulated rate — is one of the most profound mathematical facts, appearing in physics (work-energy theorem), economics (total revenue from marginal revenue), and probability (CDF from density).

Systematic Integration Strategy

Faced with any integral, use this decision tree. Step 1: Does the integrand match a basic formula directly (power, exponential, trig, inverse trig)? If yes, apply the formula. Step 2: Is there a composite function where the inner function's derivative also appears (possibly with a constant factor)? If yes, use u-substitution. Step 3: Is the integrand a product of two "unrelated" functions (polynomial × exponential, polynomial × trig, log × anything)? Try integration by parts. Step 4: Is the integrand a rational function (polynomial / polynomial)? Use partial fraction decomposition after long division if necessary. Step 5: Does it involve √(a²−x²), √(a²+x²), or √(x²−a²)? Use trigonometric substitution.

Numerical Integration — When Closed Forms Don't Exist

Not every integral has a closed-form antiderivative. e^(−x²), sin(x)/x, 1/ln(x), and √(1+x⁴) are proved to have no antiderivative in terms of elementary functions. For these, numerical methods are required. The Trapezoidal Rule approximates the integral as a sum of trapezoids: ∫ₐᵇf dx ≈ (h/2)[f(x₀)+2f(x₁)+···+2f(xₙ₋₁)+f(xₙ)] where h=(b−a)/n. Simpson's Rule uses parabolic arcs instead of straight lines and achieves much higher accuracy for the same number of function evaluations. Gaussian quadrature chooses evaluation points optimally and is exact for polynomials up to degree 2n−1 using only n points — the method used in scientific computing and finite element software.