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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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