Key Idea: Integration reverses differentiation. You use it to get a quantity from its rate, like the volume in a tank from its fill rate, and a given condition fixes the constant C.
Integration techniques
- Indefinite integral — Reverse the derivative, always + C.
- Reverse power rule — Raise the power by 1, divide by the new power.
- Reverse chain — Spot an inside g(x) with g′(x) alongside. Divide by it when linear; otherwise set u = g(x).
- Check — Differentiate your answer.
| Integrand | Result |
|---|---|
| xⁿ | xⁿ⁺¹/(n+1) + C |
| 1/x and 1/(ax+b) | ln (÷ a) |
| Inside² or trickier | Set u = inside |
A condition such as V(0) = 5 turns a family of curves into one answer.
Rate R(t) = 1/(t + 2), with 5 litres at t = 0. Find V(t). V(t) = ln((t + 2)/2) + 5.
Find ∫ x·eˣ² dx. ½eˣ² + C.
Find ∫ x³ dx. x⁴/4 + C.
How do you check ½eˣ² is right? Differentiate it. You get x·eˣ² back.