Integration
Reverse of differentiation. ∫ xn dx = xn+1/(n+1) + c (n ≠ −1).
Y12 polynomial integration. Y13 adds substitution, parts, exponential and trig integrands. Definite integral ∫ab f(x) dx = F(b) − F(a) gives the signed area between curve and x-axis.
Worked examples
∫ (3x² + 2) dx = x³ + 2x + c.
∫02 x² dx = [x³/3] from 0 to 2 = 8/3.
Area below x-axis is negative; sum of magnitudes for total area.
Frequently asked questions
Why +c?
Constant of integration. Differentiation loses the constant; integration recovers it as ‘some unknown constant’.
∫ 1/x dx?
= ln|x| + c. Special case — the n=−1 power rule fails.