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← Unit 6: Integration and Accumulation of Change

Antiderivatives and u-Substitution

Antidifferentiation reverses differentiation. Basic antiderivative rules mirror derivative rules in reverse; u-substitution reverses the chain rule to handle composite integrands.

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Basic antiderivative rules

∫xⁿ dx = x^(n+1)/(n+1) + C (for n≠-1). ∫1/x dx = ln|x| + C. ∫eˣ dx = eˣ + C. ∫cos(x)dx = sin(x)+C, ∫sin(x)dx = -cos(x)+C. Always add '+C' for an indefinite integral — antiderivatives are only unique up to a constant.

u-substitution

u-substitution reverses the chain rule. Look for a piece of the integrand (call it u) whose derivative (or a constant multiple of it) also appears in the integrand. Substitute, rewrite the whole integral in terms of u, integrate, then substitute back.

Worked Example

Evaluate ∫ 2x·cos(x²) dx.

  1. Let u = x², so du = 2x dx — and 2x dx appears exactly in the integral already.
  2. Rewrite: ∫ cos(u) du.
  3. Integrate: sin(u) + C.
  4. Substitute back u=x²: sin(x²) + C.

Answer: sin(x²) + C

Key terms (2)
Antiderivative
A function F such that F'(x) = f(x); always includes '+C' for the indefinite integral.
u-substitution
A technique reversing the chain rule by substituting u for an inner function whose derivative also appears in the integrand.

Practice Quiz

Question 1 of 3Score so far: 0/0

∫ x⁴ dx = ?