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Unit 6: Integration & Accumulation of Change

The FTC & U-Substitution

Applying the Fundamental Theorem of Calculus and finding antiderivatives with u-substitution.

Advanced25 min lesson2 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

  • Riemann Sums & the Definite Integral

The Fundamental Theorem, and Reversing the Chain Rule

Definition — The Fundamental Theorem of Calculus

Part 1: for A(x) = ∫ₐˣf(t)dt, A'(x) = f(x) — a thin sliver of added area, f(x)·h, divided by h, becomes exact as h → 0. Part 2: if F' = f, then ∫ₐᵇf(x)dx = F(b) − F(a), evaluating a definite integral directly from any antiderivative.
∫abf(x) dx=F(b)−F(a),where F′=f\int_a^b f(x)\,dx = F(b) - F(a), \quad \text{where } F' = f

Worked Example — Evaluating a Definite Integral

Evaluate ∫₁³(4x³ − 2x)dx. An antiderivative is F(x) = x⁴ − x² (check: d/dx[x⁴−x²] = 4x³−2x). F(3) − F(1) = (81−9) − (1−1) = 72 − 0 = 72.

Worked Example — An Accumulation Function with a Non-x Upper Bound

Let A(x) = ∫₃ˣ(t²−1)dt; by FTC Part 1, A'(x) = x²−1. Now let B(x) = ∫₃^(x³)(t²−1)dt — the upper bound is x³. Since B(x) = A(x³), the chain rule gives B'(x) = A'(x³)·3x² = (x⁶−1)(3x²) = 3x⁸ − 3x².

Definition — U-Substitution

Reversing the chain rule for integration: pick u equal to an inner function whose derivative (du) appears elsewhere in the integral, rewrite entirely in u, integrate, then substitute back.

Worked Example — Finding an Antiderivative by Substitution

Evaluate ∫3x²(x³+2)⁴dx. Let u = x³+2, du = 3x²dx — matching exactly. ∫u⁴du = u⁵/5 + C = (x³+2)⁵/5 + C. Check: d/dx[(x³+2)⁵/5] = (5)(x³+2)⁴(3x²)/5 = 3x²(x³+2)⁴ — matching.

Tip

A correct substitution leaves no x behind — every remaining piece of the integral must be expressible purely in u.

Common Mistakes

  • Forgetting the chain rule factor when an accumulation function's upper bound is a function of x rather than x itself.

    FTC Part 1 alone only handles a plain x upper bound — a bound like x³ needs an extra factor, its own derivative, multiplied in.

Key Takeaways

  • FTC Part 1 connects an accumulation function's derivative back to the original function; Part 2 evaluates a definite integral from any antiderivative.
  • A non-x upper bound on an accumulation function needs an extra chain rule factor.
  • U-substitution reverses the chain rule, rewriting an integral entirely in terms of a well-chosen inner function.

Summary

This closes Unit 6. Unit 7 turns to equations that specify a rate of change directly, and solves them using integration.

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