Integral Calculator
Compute integrals of power functions instantly. Choose indefinite integration to find the antiderivative or definite integration to calculate the area under a curve.
Understanding Integration
Integration is one of the two fundamental operations in calculus, alongside differentiation. While derivatives measure rates of change, integrals measure accumulation. If you know how fast a car is traveling at every moment, integration tells you the total distance traveled.
The indefinite integral, written ∫f(x)dx, finds all functions whose derivative equals f(x). These functions are called antiderivatives. Because constants disappear when you differentiate, every antiderivative differs by at most a constant, which is why we write + C.
The definite integral, written ∫[a to b]f(x)dx, calculates a specific numerical value by evaluating the antiderivative at the upper bound b, evaluating it at the lower bound a, then subtracting. This process is called the Fundamental Theorem of Calculus.
The Power Rule for Integration
Just as there's a power rule for derivatives, there's one for integrals. To integrate ax^n, add 1 to the exponent and divide the coefficient by the new exponent: ∫ax^n dx = a·x^(n+1)/(n+1) + C.
For example, ∫3x² dx becomes 3·x³/3 + C = x³ + C. The coefficient 3 gets divided by the new exponent 3, which cancels. For ∫5x⁴ dx, you get 5·x⁵/5 + C = x⁵ + C.
The rule works for negative and fractional exponents too. The integral of x^(-2) is x^(-1)/(-1) + C = -1/x + C. The only exception is n = -1, which produces a natural logarithm instead: ∫x^(-1) dx = ln|x| + C.
Real-World Applications
Integration solves countless practical problems. Engineers use it to calculate volumes of revolution, moments of inertia, and centers of mass. Physicists integrate acceleration to find velocity, and velocity to find position. Economists integrate marginal functions to find total cost, revenue, and profit.
In probability, integrals compute cumulative distribution functions. In signal processing, convolution integrals combine signals. In computer graphics, path integrals simulate realistic lighting by tracing billions of light rays.
Even simple applications are powerful. The area under a velocity-time graph gives distance traveled. The area under a power-time graph gives energy consumed. The area under a demand curve above a price line gives consumer surplus. Any time you need to add up infinitely many infinitesimal contributions, integration is the tool you need.
Frequently Asked Questions
What is an integral?
An integral is the reverse of a derivative. It represents the accumulated total of a function over an interval. Geometrically, it measures the area under a curve.
What is the difference between definite and indefinite integrals?
An indefinite integral gives a family of functions (the antiderivative plus a constant C). A definite integral evaluates between two bounds and produces a specific number.
What is the power rule for integration?
To integrate x^n, add 1 to the exponent and divide by the new exponent: ∫x^n dx = x^(n+1)/(n+1) + C. This works for all n except n = -1.
Why does the indefinite integral include + C?
The constant C represents any constant value, because the derivative of a constant is zero. Without C, you're missing infinitely many valid antiderivatives.
What happens when integrating x^(-1)?
The integral of x^(-1) = 1/x is a special case: ∫(1/x)dx = ln|x| + C. This is because the power rule fails when n = -1 (you'd divide by zero).