| Version 7 |
Version 6 |
| Let $f\colon[a,b]\to \mathbf R$ be a continuous function, let $c\in[a,b]$ be given |
Let $f\colon[a,b]\to \mathbf R$ be a continuous function, let $c\in[a,b]$ be given |
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and consider the integral function $F$ defined on $[a,b]$ as
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and consider the integral function |
| F(x)= \int_c^x f(t)\, dt. |
F(x)= \int_c^x f(t)\, dt. |
| Then $F$ is an antiderivative of $f$ that is, |
Then $F$ is an antiderivative of $f$ that is, |
| $F$ is differentiable in $[a,b]$ and |
$F$ is differentiable in $[a,b]$ and |
| F'(x)=f(x)\qquad \forall x\in [a,b]. |
F'(x)=f(x)\qquad \forall x\in [a,b]. |
| The previous relation rewritten as |
The previous relation rewritten as |
| \frac{d}{dx} \int_c^x f(t)\, dt = f(x) |
\frac{d}{dx} \int_c^x f(t)\, dt = f(x) |
| shows that the differentiation operator $\frac{d}{dx}$ is the inverse of the integration operator $\int_c^x$. |
shows that the differentiation operator $\frac{d}{dx}$ is the inverse of the integration operator $\int_c^x$. |
| On the other hand if $f\colon[a,b]\to \mathbf R$ is a continuous function |
On the other hand if $f\colon[a,b]\to \mathbf R$ is a continuous function |
| and $G\colon[a,b]\to \mathbf R$ is any antiderivative of $f$, i.e.\ $G'(x)=f(x)$ for all $x\in[a,b]$, then |
and $G\colon[a,b]\to \mathbf R$ is any antiderivative of $f$, i.e.\ $G'(x)=f(x)$ for all $x\in[a,b]$, then |
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\int_a^b f(t) \, dt = \, dt = G(b)-G(a).
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\int_a^b f(t) \, dt = \int_a^b G'(t)\, dt = G(b)-G(a).
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| This shows that up to a constant, the integration operator is the inverse of the derivative operator: |
This shows that up to a constant, the integration operator is the inverse of the derivative operator: |
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\int_a^x D G = G - G(a).
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\int_a^x D f = f - f(a).
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| \] |
\] |
