integral from 0 to 10 of 1/(100+x^2)

Lösung

\int_{0}^{10} \frac{1}{100 + x ^{2}} d x

\frac{π}{40}

Dezimale

0.07853 \dots

Schritte zur Lösung

\int_{0}^{10} \frac{1}{100 + x ^{2}} d x

Wende integrale Substitution an

= \int_{0}^{1} \frac{1}{10 ( u ^{2} + 1 )} d u

Entferne die Konstante:

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

= \frac{1}{10} \cdot \int_{0}^{1} \frac{1}{u ^{2} + 1} d u

Nutze das gemeinsame Integral :

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

= \frac{1}{10} [arctan (u)]_{0}^{1}

Berechne die Grenzen:

\frac{π}{4}

= \frac{1}{10} \cdot \frac{π}{4}

Vereinfache

= \frac{π}{40}

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