Integration runs differentiation backwards: Integration is differentiation run backwards, and every indefinite integral carries a + C.
What integration does
- Picture a car's speedometer: differentiating distance gives speed.
- Integration runs the other way — given the speed, it rebuilds the distance.
- Many curves have the same slope everywhere; they differ only by a constant height.
- So 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: Reverse-chain integration spots an inside function g(x) sitting next to its derivative g′(x).
How the pattern works
- 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 divide when that inside is linear; otherwise 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. ✓