← Integration methods
6 u-Substitution: what it is for (first principles)
Watch: u-Substitution: what it is for (first principles) Opens a YouTube search: pick a video that clicks for you You already know how to integrate simple things like ∫x³ dx . But what about ∫2x(x²+1)⁵ dx ? Expanding (x²+1)⁵ would be brutal. u-substitution is the escape hatch. The core idea: u-substitution is the chain rule run backwards. Recall differentiating a composite function: Chain rule (differentiating) \frac{d}{dx}\bigl[f(g(x))\bigr]=f'\bigl(g(x)\bigr)\cdot g'(x) Derivative of the outside × derivative of the inside. So if an integrand looks like f′(g(x)) · g′(x) , that is, some inner function AND its own derivative multiplied alongside, then integrating it just undoes the chain rule: Substitution rule (integrating) \int f'\bigl(g(x)\bigr)\cdot g'(x)\,dx=f\bigl(g(x)\bigr)+c Setting u = g(x) turns this into the much friendlier ∫ f′(u) du.The mechanics: why "du" works
- Let u = g(x) (the messy inner function).
- Differentiate it: du/dx = g′(x).
- Rearrange informally to du = g′(x) dx. This is the swap that lets you replace the whole g′(x) dx chunk with a single du .
- Now the integral is entirely in terms of u. Integrate normally, then substitute x back in at the end.