Ableitung und Integration

Verschiedene Regeln und Hilfen

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Philipp Stark

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Flashcards	16
Language	Deutsch
Category	Maths
Level	University
Created / Updated	10.06.2020 / 10.06.2020
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Partielle Integration:

\(\int (f' \cdot g) dx\)

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\(\int f' \cdot g\ dx = f \cdot g - \int f \cdot g'\ dx \ + C\)

\(\int x^{a}dx\)

integrieren

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\({\frac{1}{\alpha + 1} x^{\alpha + 1}} + C\)

als integral schreiben

\(\int\frac{1}{x} dx\)

integrieren

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\(\text{ln}(|x|) + C\)

als integral schreiben

\(\int e^x dx\)

integrieren

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\(e^x + C\)

als integral schreiben

\(\int \text{sin}(x) \ dx\)

integrieren

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\(-\text{cos}(x) + C\)

als integral schreiben

\(\int \frac{1}{\sqrt{1-x^2}} dx\)

integrieren

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\(\text{arcsin}(x) + C\)

als Integral schreiben

\(\int \frac{1}{1+x^2} dx\)

integrieren

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\(\text{arctan}(x) + C\)

als Integral schreiben

\(\int \text{sinh}(x) \ dx\)

integrieren

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\(\text{cosh}(x) + C\)

(sinh' = cosh; cosh' = sinh; ohne Vorzeichenänderung)

als Integral schreiben

\(\int \frac{-1}{\sqrt{1-x^2}} dx\)

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\(\int \frac{-1}{\sqrt{1-x^2}} dx=-\text{arcsin}(x) + C_1 = \text{arccos}(x) + C_2\)

( \(\text{arcsin}(x) = -\text{arccos}(x)+\frac{\pi}{2} \) )

Kettenregel (Ableitung)

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\((\ f(g(x))\ )' = f'(g(x)) \cdot g'(x)\)

Kettenregel (Integration)

\(\int f'(g(x)) \cdot g'(x) \ dx = f(g(x))\)

Substitution beim Integrieren

\(\int_{a}^{b} f(g(x))\ dx\)

Zu substituierenden Term bestimmen \(\longrightarrow g(x)=y \implies x = g^{-1}(y)\)
dx berechnen \(\frac{dx}{dy}=\frac{d}{dy}g^{-1}(y) = (g^{-1})'(y) \ \ \implies\ \ dx = (g^{-1})'(y)\cdot dy\)
Neue Integrationsgrenzen bestimmen \(a_y=g(a),\ b_y=g(b)\)
Integral ausrechnen und Rücksubstitution
\(\int_{a_y}^{b_y} f(y) \cdot (g^{-1})'(y)\ dy = h(a_y,b_y) + C = h(g(a),g(b)) + C\)

\({\int \frac{1}{x\ \text{log}(x)}dx}\)

integriere

\(\int \frac{1}{\text{log}(x)}\frac{1}{x} dx = \text{log(log}(x)) + C\)

(Kettenregel und \(\frac{d}{dx}log(x) = \frac{1}{x}\) )

\(\int \text{tan}(x) dx\)

integriere

\(\int \text{tan}(x) dx = \int \frac{\text{sin}(x)}{\text{cos}(x)} dx = -\int \frac{-\text{sin}(x)}{\text{cos}(x)} dx = \\-\int \frac{1}{\text{cos}(x)}(-\text{sin}(x))\ dx =-\text{log(cos}(x)) +C\)

\(\int_{1}^{2} (x^2 + 1)\text{log}(x) dx\)

integriere

\(\underbrace{\int_{1}^{2} x^2 \text{log}(x) dx }_{[\frac{x^3}{3}\text{log}(x)]_1^2 - \int_{1}^{2} \frac{1}{3}x^3 \cdot \frac{1}{x} dx}+ \underbrace{\int_{1}^{2} 1 \text{log}(x) dx }_{[x\text{log}(x)]_1^2 - \int_1^2 x\frac{1}{x} dx}\)
\(= [\frac{x^3}{3}\text{log}(x)- \frac{x^3}{9}+ x\text{log}(x)- x]_1^2\)\(= [(\frac{x^3}{3}+x)\text{log}(x)- \frac{x^3}{9}- x]_1^2\)
\(\dots\)

\(\int \frac{f'(x)}{f(x)} dx\)

integrieren

\(\text{ln}(|f(x)|) + C\)

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Ableitung und Integration

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