How to Use the Basic Integral Formula Calculator
Integration is the reverse of differentiation — it finds the area under a curve or recovers the original function from its rate of change. This calculator applies the basic integral formulas for polynomial, trigonometric, and exponential functions directly, computing both the indefinite integral (F(x)+C) and the definite integral over the interval you specify.
For a polynomial a·x^n, the indefinite integral uses the power rule: raise the exponent by 1 and divide by the new exponent (a/(n+1)·x^(n+1)). This formula breaks down when n equals -1, since that case actually integrates to a logarithm — so it isn't supported here. Trigonometric and exponential functions use the standard formulas for sin, cos, and e^x.
When the exponent is negative or not a whole number, the polynomial itself may be undefined near x=0, so in that case the integration interval must be entirely greater than 0.
Frequently Asked Questions
Yes. If the function dips below the x-axis over part of the interval, the definite integral can be negative — it means that portion of the area lies below the x-axis.
The power rule divides by the exponent plus 1, and when the exponent is -1 that denominator becomes 0. That specific case integrates to a logarithm (ln|x|) instead, which needs a separate formula.