Integration by Substitution

This lesson covers integration by substitution (also called u-substitution), a powerful technique for evaluating integrals that cannot be found by simple inspection. It is the reverse of the chain rule for differentiation and is a key topic in A-Level Mathematics.

---

The Idea

If we can write an integral in the form:

∫ f(g(x)) × g'(x) dx

then by substituting u = g(x), du = g'(x) dx, the integral simplifies to:

∫ f(u) du

which is often much easier to integrate.

---

Method for Indefinite Integrals

1. Choose a substitution u = g(x).
2. Find du/dx and rearrange to express dx in terms of du.
3. Replace all x terms in the integral with u terms.
4. Integrate with respect to u.
5. Substitute back to express the answer in terms of x.

---

Example 1: Basic Substitution

Find ∫ 2x(x² + 3)⁴ dx.

Let u = x² + 3. Then du/dx = 2x, so du = 2x dx.

∫ 2x(x² + 3)⁴ dx = ∫ u⁴ du = u⁵/5 + C = (x² + 3)⁵/5 + C

---

Example 2: Adjusting the Constant