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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:

$$\int_a^bf(\phi(t))\phi’(t)dt=\int_{\phi(a)}^{\phi(b)}f(x)dx.$$

Mnemonic Notation

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

$$\int_a^bf(\phi(t))d\phi(t)=\int_{\phi(a)}^{\phi(b)}f(x)dx,$$

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 $\phi(b)$.

| | | | | created: 2017-08-03 22:46:02 | modified: 2017-08-03 23:20:56 | by: bookofproofs | references: [581]

1.Proof: (related to "Integration by Substitution")

Edit or AddNotationAxiomatic Method

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Bibliography (further reading)

[581] Forster Otto: “Analysis 1, Differential- und Integralrechnung einer Veränderlichen”, Vieweg Studium, 1983