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Rechenregeln zum Integral

Rechenregeln:

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x

Damit folgt sofort:

\int_{a}^{a} f (x) d x = 0

Beispiel

\int_{0}^{1} 2 x d x = - \int_{1}^{0} 2 x d x,

denn:

\int_{0}^{1} 2 x d x = {[x^{2}]}_{0}^{1} = 1 - 0 = - (0 - 1) = - {[x^{2}]}_{1}^{0} = - \int_{1}^{0} 2 x d x

\int_{a}^{b} f (x) d x + \int_{b}^{c} f (x) d x = \int_{a}^{c} f (x) d x

Beispiel

\int_{0}^{1} 2 x d x + \int_{1}^{2} 2 x d x = \int_{0}^{2} 2 x d x,

denn:

\int_{0}^{1} 2 x d x + \int_{1}^{2} 2 x d x = {[x^{2}]}_{0}^{1} + {[x^{2}]}_{1}^{2} = (1 - 0) + (4 - 1) = 4 - 0 = {[x^{2}]}_{0}^{2} = \int_{0}^{2} 2 x d x

\int_{a}^{b} c \cdot f (x) d x = c \cdot \int_{a}^{b} f (x) d x, c =

Konstante

Beispiel

3 \cdot \int_{1}^{2} x^{2} = \int_{1}^{2} 3 \cdot x^{2} d x,

denn:

3 \cdot \int_{1}^{2} x^{2} = 3 \cdot {[\frac{1}{3} x^{3}]}_{1}^{2} = 3 \cdot (\frac{1}{3} \cdot 8 - \frac{1}{3}) = 8 - 1 = {[x^{3}]}_{1}^{2} = \int_{1}^{2} 3 \cdot x^{2} d x

\int_{a}^{b} (f (x) \pm g (x)) d x = \int_{a}^{b} f (x) d x \pm \int_{a}^{b} g (x) d x

Beispiel

\int_{0}^{1} (2 x + 1) d x = \int_{0}^{1} 2 x d x + \int_{0}^{1} 1 d x,

denn:

\int_{0}^{1} (2 x + 1) d x = {[x^{2} + x]}_{0}^{1} = (1^{2} + 1) - (0^{2} + 0) = (1^{2} - 0^{2}) + (1 - 0) = {[x^{2}]}_{0}^{1} + {[x]}_{0}^{1} = \int_{0}^{1} 2 x d x + \int_{0}^{1} 1 d x

| \int_{a}^{b} f (x) d x | \leq \int_{a}^{b} | f (x) | d x, (b > a)

Beispiel

| \int_{- 1}^{1} x d x | < \int_{- 1}^{1} | x | d x,

denn:

| \int_{- 1}^{1} x d x | = | {[\frac{1}{2} x^{2}]}_{- 1}^{1} | = | \frac{1}{2} - \frac{1}{2} | = 0

und

\int_{- 1}^{1} | x | d x = \int_{- 1}^{0} - x d x + \int_{0}^{1} x d x = {[- \frac{1}{2} x^{2}]}_{- 1}^{0} + {[\frac{1}{2} x^{2}]}_{0}^{1} = (0 - (- \frac{1}{2})) + (\frac{1}{2} + 0) = \frac{1}{2} + \frac{1}{2} = 1

\Rightarrow 0 < 1

Verknüpfte Inhalte

Kategorie: Bestimmtes Integral

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