integral from-a to a of a/(pi(x^2+a^2))

Lösung

\int_{- a}^{a} \frac{a}{π ( x ^{2} + a ^{2} )} d x

\frac{1}{2}

Dezimale

0.5

Schritte zur Lösung

\int_{- a}^{a} \frac{a}{π ( x ^{2} + a ^{2} )} d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{a}{π} \cdot \int_{- a}^{a} \frac{1}{x ^{2} + a ^{2}} d x

Wende integrale Substitution an

= \frac{a}{π} \cdot \int_{- 1}^{1} \frac{1}{a ( u ^{2} + 1 )} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{a}{π} \cdot \frac{1}{a} \cdot \int_{- 1}^{1} \frac{1}{u ^{2} + 1} d u

Nutze das gemeinsame Integral :

\int \frac{1}{u ^{2} + 1} d u = arctan (u)

= \frac{a}{π} \cdot \frac{1}{a} [arctan (u)]_{- 1}^{1}

Vereinfache

\frac{a}{π} \cdot \frac{1}{a} [arctan (u)]_{- 1}^{1} : \frac{1}{π} [arctan (u)]_{- 1}^{1}

= \frac{1}{π} [arctan (u)]_{- 1}^{1}

Berechne die Grenzen:

\frac{π}{2}

= \frac{1}{π} \cdot \frac{π}{2}

Vereinfache

= \frac{1}{2}

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