Advanced · 15 min
An introduction to integration
Find simple antiderivatives and calculate accumulated change.
Reverse a derivative
An antiderivative of xⁿ for a nonnegative integer n is xⁿ⁺¹/(n + 1). Add a constant C because constants disappear when differentiated.
Worked example
An antiderivative of 2x is x² + C, since the derivative of x² + C is 2x.
Evaluate between limits
A definite integral from a to b equals F(b) − F(a) for an antiderivative F of a continuous integrand. It gives signed accumulation; below-axis contributions are negative.
Worked example
The integral of 2x from 1 to 3 is 3² − 1² = 8.
Check integration by differentiation
- Find an antiderivative of 3x² and differentiate it to check.
- Use it to calculate the definite integral from x = 0 to x = 2.