Integral Calculator

What Is an Integral?

An integral calculates the total accumulation of a quantity. If the derivative tells you the rate of change at each point, the integral tells you the total change over an interval. Geometrically, the definite integral of a function from a to b equals the area under the curve between those points. The integral is the inverse operation of differentiation, which is why it is also called an antiderivative. Enter any function in the calculator above to find its integral with step-by-step explanation.

Definite vs Indefinite Integrals

An indefinite integral finds the general antiderivative: the integral of 2x dx = x² + C, where C is the constant of integration (since the derivative of any constant is 0). A definite integral evaluates the antiderivative between two limits: the integral of 2x from 0 to 3 = [x²] from 0 to 3 = 9 - 0 = 9. The definite integral gives a specific numerical answer representing the signed area under the curve. The indefinite integral gives a family of functions that all have the same derivative.

Basic Integration Rules

Power rule: The integral of x^n is x^(n+1)/(n+1) + C (for n not equal to -1). The integral of x² is x³/3 + C. The integral of x^4 is x^5/5 + C. Constant multiple: Pull constants out. The integral of 5x² is 5 times x³/3 = 5x³/3 + C. Sum/difference: Integrate term by term. The integral of (3x² + 2x - 1) is x³ + x² - x + C. Special cases: The integral of 1/x is ln|x| + C. The integral of e^x is e^x + C.

Integration Techniques

Substitution (u-sub): The reverse of the chain rule. For the integral of 2x times cos(x²), let u = x², du = 2x dx, giving the integral of cos(u) du = sin(u) + C = sin(x²) + C. Integration by parts: The reverse of the product rule. The integral of u dv = uv minus the integral of v du. Used for products like x times e^x or x times sin(x). Partial fractions: Decomposes rational functions into simpler fractions that can be integrated individually. Trigonometric substitution: Replaces expressions involving square roots with trig functions. Each technique handles a different category of integrals that basic rules cannot solve directly.

What Is the Area Under a Curve?

The definite integral computes the signed area between a function and the x-axis. Area above the x-axis is positive. Area below is negative. The total area (ignoring sign) requires integrating the absolute value of the function. For f(x) = x from -2 to 2: the integral is 0 (negative area from -2 to 0 cancels positive area from 0 to 2), but the total area is 4. This distinction matters in physics (net displacement vs total distance) and economics (net profit vs total revenue).

Applications of Integrals

Physics uses integrals to calculate work (force integrated over distance), electric charge (current integrated over time), and center of mass (position weighted by density, integrated over volume). Engineering calculates structural loads by integrating pressure distributions over surfaces. Economics uses integrals for consumer and producer surplus (area between supply and demand curves). Probability theory defines continuous distributions through integrals (the area under a probability density function equals 1). Statistics uses integrals to calculate expected values and cumulative distribution functions. Any situation requiring total accumulation from a rate of change uses integration.

The Fundamental Theorem of Calculus

The fundamental theorem of calculus connects derivatives and integrals as inverse operations. Part 1 states that if F(x) is the integral of f(t) from a to x, then F'(x) = f(x). Part 2 states that the definite integral of f(x) from a to b equals F(b) - F(a), where F is any antiderivative of f. This theorem is why finding an antiderivative is the key to evaluating definite integrals. It unifies the two central operations of calculus and provides the computational method used in every integration calculation, from homework problems to engineering simulations and scientific models.

Numerical Integration

When a function has no closed-form antiderivative (like e^(-x²), the bell curve function), numerical methods approximate the integral. The trapezoidal rule approximates the area using trapezoids. Simpson's rule uses parabolic arcs for greater accuracy. Monte Carlo integration uses random sampling, effective for high-dimensional integrals. Computers use these methods to evaluate integrals in physics simulations, financial modeling, and engineering analysis where exact antiderivatives do not exist. The calculator above finds exact symbolic answers when possible and provides numerical approximations when no closed form exists.

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