What integration means
Integration is the reverse process of differentiation. If we know the derivative dy/dx = f(x), then integrating f(x) recovers the original function y. We write y = ∫ f(x) dx, read as "the integral of f(x) with respect to x".
Differentiating any constant gives 0, so when we reverse the process we cannot know which constant was there originally. Every indefinite integral therefore carries an arbitrary constant c.
Key formula
∫ xn dx = xn+1/(n+1) + c, where n ≠ −1. Also ∫ a dx = ax + c and ∫ a·f(x) dx = a ∫ f(x) dx.
Worked example
Worked example
Given dy/dx = 6x² − 4x + 5, find y. Integrate term by term: y = 6·x³/3 − 4·x²/2 + 5x + c = 2x³ − 2x² + 5x + c. Check by differentiating: d/dx(2x³ − 2x² + 5x + c) = 6x² − 4x + 5. ✓
Finding the constant c
If a boundary condition is given, substitute it to fix c. For example, if dy/dx = 3x² and the curve passes through (1, 4): integrating gives y = x³ + c, then 4 = 1³ + c so c = 3, giving y = x³ + 3.
Remember
- Add + c to every indefinite integral.
- Increase the power by 1, then divide by the new power.
- Always check an integral by differentiating your answer.