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Concept |
Content |
Derivatives |
Derivatives of elementary functions and rules for derivatives. |
Integral |
Some properties of integrals |
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As a reminder: Chain rule for differentiation: \(F(\varphi)' = F'(\varphi)\cdot
\varphi'\)
Theorem 1 (Substitution). Let \(F\) be an antiderivative of \(f\). Then it holds \[\int f(\varphi(x))\varphi'(x)\,\mathrm{d} x =
F(\varphi(x)) + c,\] and \[\int_{a}^b
f(\varphi(x)) \varphi'(x)\,\mathrm{d} x =
\int_{\varphi(a)}^{\varphi(b)} f(t)\,\mathrm{d} t =
F(t)\bigr|_{t=\varphi(a)}^{\varphi(b)} =
F(\varphi(x))\bigr|_{x=a}^b.\]
Typical
applications
\(\int f(\alpha x+\beta)\,\mathrm{d}
x\), then \(\varphi(x) = \alpha
x+\beta\)
\(\int f(\sin(x))\cos(x)\,\mathrm{d}
x\), then \(\varphi(x) =
\sin(x)\);
analogously with \(\sin\) and \(\cos\) switched
\(\int
\frac{f'(x)}{f(x)}\,\mathrm{d} x\), then \(\varphi(x) = f(x)\)
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