Theorem: Integration by Substitution

Let $I$ be a real interval, let $f:I\to\mathbb R$ be a continuous function, and let $\phi:[a,b]\to\mathbb R$ be a continouosly differentiable function with an image contained in $I$, formally $\phi([a,b])\subset I$. The term integration by substitution refers to the Riemann integral of the function $f(\phi(t))\phi'(t)$ by the following formula:


Mnemonic Notation

The above formula can be easily memorized by setting $d\phi(t):=\phi'(x)dx$ and writing


where $x$ is simply substituted by $\phi(t).$ If $t$ runs through the values from $a$ to $b$, then $x$ runs through the values from $\phi(a)$ to $\phi(b)$.

Proofs: 1

Proofs: 1

Thank you to the contributors under CC BY-SA 4.0!




  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983