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| Concept |
Content |
| Derivatives |
Derivatives of elementary functions and rules for derivatives. |
| Integral |
Some properties of integrals |
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Antiderivative
Definition 1. For \(I\subseteq\mathbb{R}\) interval and \(f: I\to\mathbb{R}\), the function \(F: I\to\mathbb{R}\) is an
antiderivative of \(f\), if \(F\) is differentiable and \(F'=f\).
Example 2. Let \(f:
[0,1]\to\mathbb{R}\), \(f(x):=2x\). Then \(F: [0,1]\to\mathbb{R}\), \(F(x):=x^2\) is an antiderivative of \(f\).
If \(F\) is an antiderivative of
\(f\) then also \(F+c\) with any \(c\in\mathbb{R}\).
Antiderivatives are unique up to a constant.
“Antiderivatives form the reversal of derivatives.”
If \(F\) is an antiderivative of
\(f\), then it holds \[\int_a^b f(t)\,\mathrm{d} t =
F(b)-F(a)=:F(x)\bigr|_{x=a}^b.\]
Hence: The calculation of integrals is reduced to
the calculation of antiderivatives (i.e. indefinite integrals) and their
evaluation at the integral limits.
Determination of
antiderivatives
“Differentiation is mechanics, integration is art.”
| \(x^{\alpha}\) |
\(\alpha
x^{\alpha - 1}\) |
| \(\mathrm{e}^x\) |
\(\mathrm{e}^x\) |
| \(\ln
|x|\) |
\(\frac{1}{x}\) |
| \(\sin(x)\) |
\(\cos(x)\) |
| \(\cos(x)\) |
\(-\sin(x)\) |
| \(\int
f(x)\,\mathrm{d} x + c\) |
\(f(x)\) |
Linearity: \[\int
\alpha f(x) + \beta g(x)\,\mathrm{d} x = \alpha \int f(x)\,\mathrm{d} x
+ \beta \int g(x)\,\mathrm{d} x\]
Products: integration by parts \[\int f(x)g'(x)\,\mathrm{d} x = f(x) g(x) -
\int f'(x) g(x)\,\mathrm{d} x\]
(Special) quotients: partial fraction
decomposition
(Special) compositions: substitution \[\int f(g(x)) g'(x)\,\mathrm{d} x = F(g(x)) +
c\qquad\qquad\]
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