First Fundamental Theorem of Calculus
Linking differentiation and integration.
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Study First Fundamental Theorem of Calculus
The next result now states that integrals can be determined by inversion of differentiation.
Theorem 1 (Fundamental theorem of differentiation and integration). Let \(I\) be an interval and a continuous \(f:I\to\mathbb{R}\) be given. Let \(F:I\to\mathbb{R}\) be an antiderivative of \(f\). Then for all \(a,b\in I\) holds \[\int_a^bf(x)\, dx=F(b)-F(a).\] We write \[\int_a^bf(x)\, dx=\left. F(x)\right|_{x=a}^{x=b}.\]
The above result gives rise to the following notation for an antiderivative: \[\int f(x)\, dx:=F(x).\] Based on our knowledge about differentiation, we now collect some antiderivatives of important functions in the following table.
| \(f(x)\) | \(\int f(x)\, dx\) |
|---|---|
| \(\displaystyle x^n,\quad n\in\mathbb{N}\) | \(\displaystyle \frac1{n+1}x^{n+1}\) |
| \(\displaystyle x^{-1},\quad x\neq0\) | \(\displaystyle \log(|x|)\) |
| \(\displaystyle x^{-n},\quad x\neq0,n\in\mathbb{N}, n\neq1\) | \(\displaystyle \frac1{1-n}x^{1-n}\) |
| \(\displaystyle \exp(x)\) | \(\displaystyle \exp(x)\) |
| \(\displaystyle \sinh(x)\) | \(\displaystyle \cosh(x)\) |
| \(\displaystyle \cosh(x)\) | \(\displaystyle \sinh(x)\) |
| \(\displaystyle \frac1{\sqrt{1+x^2}}\) | \(\displaystyle\operatorname{arsinh}(x)\) |
| \(\displaystyle \frac1{\sqrt{x^2-1}},\quad x>1\) | \(\displaystyle\operatorname{arcosh}(x)\) |
| \(\displaystyle \frac1{1-x^2},\quad |x|<1\) | \(\displaystyle\operatorname{artanh}(x)\) |
| \(\displaystyle \sin(x)\) | \(\displaystyle-\cos(x)\) |
| \(\displaystyle \cos(x)\) | \(\displaystyle\sin(x)\) |
| \(\displaystyle \frac1{\cos^2(x)}=1+\tan^2(x)\) | \(\displaystyle \tan(x)\) |
| \(\displaystyle \frac1{\sqrt{1-x^2}},\quad|x|<1\) | \(\displaystyle \arcsin(x)\) |
| \(\displaystyle -\frac1{\sqrt{1-x^2}},\quad|x|<1\) | \(\displaystyle \arccos(x)\) |
| \(\displaystyle \frac1{1+x^2}\) | \(\displaystyle \arctan(x)\) |
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