If derivatives answer the question of rate of change, integrals answer the question of accumulation. How much total area is under a curve? How far did something travel if you know its velocity over time? What is the total charge accumulated given the current? Integration provides answers to all these questions and forms the second major pillar of calculus.



There are two closely related ideas: the indefinite integral, which is essentially the antiderivative, and the definite integral, which computes a specific numerical value. The indefinite integral of f(x) is written as the integral symbol followed by f(x)dx, and it represents the family of all functions whose derivative is f(x). The definite integral from a to b adds bounds and computes the net signed area under the curve between those limits.



The Fundamental Theorem of Calculus connects these two ideas beautifully. It states that the definite integral of f from a to b equals F(b) - F(a), where F is any antiderivative of f. This theorem is extraordinary because it converts an infinite summation problem (the area under a curve) into a simple evaluation problem. Without it, computing areas would require exhausting Riemann sum calculations.



Several techniques expand the range of integrals you can solve. Substitution, also called u-substitution, is the reverse of the chain rule and handles compositions. Integration by parts reverses the product rule: the integral of u dv equals uv minus the integral of v du. Partial fractions break rational functions into simpler terms. Trigonometric substitutions handle expressions involving square roots of quadratic forms.



Not all integrals have closed-form solutions. When you encounter these, numerical methods come to the rescue. Simpson’s rule and the trapezoidal rule approximate definite integrals by dividing the area into shapes whose areas are easy to compute. Our integral calculator uses these methods to deliver accurate results even when symbolic integration fails.



Improper integrals extend the concept to infinite intervals and functions with vertical asymptotes. An integral from 1 to infinity of 1/x^2 dx converges to 1, while the integral of 1/x dx diverges. Understanding convergence is critical in probability theory, where probability density functions must integrate to 1 over their entire domain.



Applications of integration are vast. In physics, work done by a variable force is the integral of force with respect to displacement. In probability, cumulative distribution functions are integrals of density functions. In engineering, the center of mass of an object requires integrating over its volume. In economics, consumer surplus is the area between the demand curve and the market price.



Building fluency with integrals requires mastering the basic formulas first. Know the integrals of polynomials, trigonometric functions, exponential functions, and their inverses. Then practice recognizing which technique applies to a given problem. Over time, you develop a sense for which substitution or manipulation will crack a particular integral.