Integral Calculator

Complete Guide to Integration

An integral is one of the two core concepts in calculus — the other being the derivative. While differentiation finds rates of change, integration finds accumulated quantities. Geometrically, a definite integral represents the net area under a curve between two bounds. This free integral calculator handles both indefinite integrals (antiderivatives) and definite integrals (area), with full step-by-step solutions and a dual-engine for maximum symbolic accuracy.

The Fundamental Theorem of Calculus

The bridge between differentiation and integration — the theorem that makes definite integrals computable:

Indefinite vs. Definite Integrals

Indefinite integral (antiderivative): Written as $\int f(x)\,dx$, it returns a family of functions with constant $+C$. Example: $\int 2x\,dx = x^2 + C$.

Definite integral (area): Written as $\int_a^b f(x)\,dx$, it returns a specific number — the net signed area under the curve from $x=a$ to $x=b$. Example: $\int_0^2 2x\,dx = 4$.

Core Integration Rules

Rule	Formula	Example
Power Rule	∫ xⁿ dx = xⁿ⁺¹/(n+1) + C	∫ x³ dx = x⁴/4 + C
Constant Multiple	∫ a·f(x) dx = a·∫f(x) dx	∫ 5x dx = 5x²/2 + C
Sum / Difference	∫ (u ± v) dx = ∫u dx ± ∫v dx	∫ (x+1) dx = x²/2 + x + C
Exponential	∫ eˣ dx = eˣ + C	∫ e^(2x) dx = e^(2x)/2 + C
Natural Log	∫ 1/x dx = ln|x| + C	∫ 1/(3x) dx = ln|x|/3 + C
Sine	∫ sin(x) dx = −cos(x) + C
Cosine	∫ cos(x) dx = sin(x) + C
Arctangent	∫ 1/(x²+1) dx = arctan(x) + C

Integration Techniques

U-Substitution: For integrals of the form $\int f(g(x))g'(x)\,dx$. Substitute $u = g(x)$, reducing to a simpler integral in $u$. Example: $\int 2x\cos(x^2)\,dx$ — let $u = x^2$, giving $\int \cos(u)\,du = \sin(u) + C = \sin(x^2) + C$.

Integration by Parts: For products of functions, using $\int u\,dv = uv - \int v\,du$. Ideal for $\int x e^x\,dx$, $\int x\sin(x)\,dx$, $\int \ln(x)\,dx$.

Partial Fractions: For rational functions, decompose into simpler fractions before integrating. Used for $\int \frac{1}{x^2-1}\,dx$.

What is a Riemann Sum?

Before the Fundamental Theorem, areas under curves were approximated by summing rectangles. A Riemann sum with $n$ rectangles over $[a,b]$:

Use the Visualize Area button above to see Left, Right, Midpoint, and Trapezoidal approximations converge to the exact value as $n$ increases.

Real-World Applications

• Physics: Displacement from velocity ($\int v(t)\,dt = s(t)$), work from force ($W = \int F\,dx$), electric charge from current.
• Engineering: Center of mass, moment of inertia, fluid pressure on a curved surface.
• Economics: Consumer surplus ($\int_0^{q^*}[D(q)-p^*]\,dq$), total revenue from marginal revenue.
• Biology: Population growth models, blood flow through arteries, drug concentration over time.
• Probability: Expected values ($E[X] = \int x f(x)\,dx$), cumulative distribution functions.

LSI Keywords: Related Concepts

This integral calculator covers: antiderivative calculator, indefinite integral solver, definite integral calculator, integration by parts calculator, u-substitution solver, area under curve calculator, Riemann sum calculator, fundamental theorem of calculus, integration rules, calculus integrator, integrate online with steps. Whether you're solving AP Calculus AB/BC problems, Calculus I–III coursework, or verifying engineering calculations, our dual-engine delivers verified results.

Frequently Asked Questions

External Resources

For deeper study, refer to OpenStax Calculus Volume 1 — Integration and Wolfram MathWorld on Integrals. Our solutions are cross-referenced against Stewart's Calculus (8th ed.) and Thomas' Calculus as part of our 4-layer verification methodology.