Basic integration rules — the reverse of basic derivatives.
Drag the right edge
y = 3·x·x + 2·x + 1area [0, 2] ≈ 14
Try it
∫ (3x² + 2x + 1) dx
Integrate term by term: x³ + x² + x + C. The shaded area above is exactly this antiderivative evaluated from 0 to the draggable edge.
Handy extras beyond the power rule
- Constant multiple: ∫ k·f(x) dx = k ∫ f(x) dx.
- ∫ 1/x dx = ln|x| + C — the n = −1 case the power rule can't touch.
- ∫ eˣ dx = eˣ + C — the function that is its own integral.
- ∫ cos x dx = sin x + C, ∫ sin x dx = −cos x + C.
Watch out
There is no product rule or quotient rule for integrals. ∫ f·g dx is *not* (∫f)(∫g). Products need substitution or integration by parts instead.
Your turn
∫ (4x³ − 6x) dx = ?
Three core rules
- Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1.
- Constant: ∫k dx = kx + C.
- Sum: ∫(f + g) dx = ∫f dx + ∫g dx.