Foundations — U-substitution — technique, change of limits for definite integrals

Before you touch a single u-substitution, you must be fluent in the alphabet the parent note speaks. Below is every symbol and idea that note assumes, built from nothing, each one leaning on the one before it.

1. Functions and the notation $f(x)$, $g(x)$

Picture a box with an input arrow and an output arrow. You drop $x=3$ in the top, a number falls out the bottom.

Why the topic needs this: u-substitution constantly writes things like $f(g(x))$ — a machine inside another machine. If "function" is fuzzy, "function of a function" is hopeless.

Look at the figure: the number enters the inner box $g$, its answer travels along the red arrow into the outer box $f$. That red arrow — the middle value $g(x)$ — is the thing u-substitution will nickname "$u$".

2. The graph and the idea of area under a curve

The red strip in the figure is one such rectangle. This single mental image explains every integral symbol that follows.

3. The symbols $dx$, $\int$, and limits $a$ to $b$

Why the topic needs this: u-substitution changes the running variable from $x$ to $u$. When it does, all three parts must be re-expressed — the height, the width $dx$, and the endpoints $a,b$. If you don't know what each part means, you won't know it needs updating. That single fact is the whole reason "change the limits" exists.

4. The derivative $g^{\prime }(x)$ and $\frac{du}{dx}$ — steepness

The red line in the figure just kisses the curve at one point — its steepness is $g^{\prime }(x)$.

5. The chain rule — the thing u-sub reverses

Why the topic needs this: the parent note's central claim is "integration is the reverse of the chain rule." The tell-tale shape $f(g(x))\cdot g^{\prime }(x)$ — something composed, multiplied by the derivative of its inside — is exactly what the chain rule produces. Spotting that shape is how you know u-substitution will work. See Chain Rule.

6. The antiderivative and $+C$

Why the topic needs this: an indefinite integral (no limits) ends in $+C$; a definite one (with limits $a,b$) gives a plain number and no $C$. U-substitution's final step differs between these two cases, so you must know which is which. See Antiderivatives & Indefinite Integrals.

7. The Fundamental Theorem — why limits give a number

Why the topic needs this: this is the law that permits limit-changing. Because the final answer is just "value at the top minus value at the bottom," it does not matter whether we plug in $x$-endpoints into $F(x)$ or $u$-endpoints into the $u$-antiderivative — as long as top and bottom describe the same physical points. That equivalence is the entire justification for the parent's "change the limits" rule. See Definite Integral & Fundamental Theorem of Calculus.

8. Linearity — pulling constants out

Why the topic needs this: Example 2 in the parent writes $xdx=\frac{1}{2}du$ — legal because $\frac{1}{2}$ is a constant. The first "steel-manned mistake" is a student illegally yanking a variable $\frac{1}{x}$ outside. Linearity is the exact rule that draws the line between the two.

Prerequisite map

Equipment checklist