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Beliebt Rechnen >

integral from sqrt(3) to 3 of 8/(9+x^2)

Lösung

\int_{3}^{3} \frac{8}{9 + x ^{2}} d x

Lösung

\frac{2 π}{9}

+1

Dezimale

0.69813 \dots

Schritte zur Lösung

\int_{3}^{3} \frac{8}{9 + x ^{2}} d x

Entferne die Konstante:

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

= 8 \cdot \int_{3}^{3} \frac{1}{9 + x ^{2}} d x

Wende integrale Substitution an

= 8 \cdot \int_{\frac{1}{3}}^{1} \frac{1}{3 ( u ^{2} + 1 )} d u

Entferne die Konstante:

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

= 8 \cdot \frac{1}{3} \cdot \int_{\frac{1}{3}}^{1} \frac{1}{u ^{2} + 1} d u

Nutze das gemeinsame Integral :

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

= 8 \cdot \frac{1}{3} [arctan (u)]_{\frac{1}{3}}^{1}

Vereinfache

8 \cdot \frac{1}{3} [arctan (u)]_{\frac{1}{3}}^{1} : \frac{8}{3} [arctan (u)]_{\frac{1}{3}}^{1}

= \frac{8}{3} [arctan (u)]_{\frac{1}{3}}^{1}

Berechne die Grenzen:

\frac{π}{12}

= \frac{8}{3} \cdot \frac{π}{12}

Vereinfache

= \frac{2 π}{9}

Graph

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