Integration runs differentiation backwards: Picture a car's speedometer: differentiating distance gives speed. Integration runs the other way — given the speed, it rebuilds the distance.
Because many curves have the same slope everywhere (they differ only by a constant height), an indefinite integral always carries a + C, the constant of integration.
The basic rule just bumps the power up by one and divides:
The one exception, and the exponential: When n = −1 the rule would divide by zero, so that single case uses a logarithm instead. You also reverse the exponential:
IB-style question — basic indefinite integral
A drone's vertical velocity is v(t) = 6t² − 4t + 2 (m s⁻¹).
Find the general expression for its height h(t).
Step by step
- Height is the integral of velocity, so integrate each term with the reverse power rule.
- Raise each power by one and divide by the new power.
- Simplify the coefficients.
Final answer
h(t) = 2t³ − 2t² + 2t + C. The + C is the unknown starting height — needed because every height curve with this velocity is just a vertical shift of the others.
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Spot the inside function and its derivative: The chain rule says the derivative of a composite has the inside's derivative multiplied on the outside. Reverse-chain integration looks for exactly that pattern: an inside function g(x) tucked inside, with (a multiple of) its derivative g′(x) sitting alongside.
When you see it, integrate the outside and divide by the derivative of the inside (only when that inside is linear, or fix the constant by substitution).
IB-style question — reverse chain into a logarithm
A pollutant concentration changes at a rate C′(x) = 8/(2x + 3).
Find the indefinite integral ∫ 8/(2x + 3) dx.
Step by step
- The inside is 2x + 3 with derivative 2, and the power is −1, so the integral is a logarithm divided by that derivative.
- Simplify the constant.
Final answer
4 ln|2x + 3| + C. Dividing by the inside's derivative (2) is the reverse-chain correction.
Substitution makes it watertight: When the inside is not linear (e.g. x² inside), use a substitution. Let u = inside, so du = g′(x) dx, rewrite the whole integral in u, integrate, then swap x back. The reverse chain is just substitution done in your head.
IB-style question — substitution with u = x²
Find ∫ x·ex² dx by substitution.
Step by step
- Let u be the inside function and differentiate it.
- Rewrite the integral entirely in u (the x·dx becomes ½ du).
- Integrate, then put u = x² back.
Final answer
½ ex² + C. Check by differentiating: ½·ex²·2x = x·ex². ✓