integral from 0 to 2 of (32)/(x^2+4)

Lösung

\int_{0}^{2} \frac{32}{x ^{2} + 4} d x

4 π

Dezimale

12.56637 \dots

Schritte zur Lösung

\int_{0}^{2} \frac{32}{x ^{2} + 4} d x

Entferne die Konstante:

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

= 32 \cdot \int_{0}^{2} \frac{1}{x ^{2} + 4} d x

Wende integrale Substitution an

= 32 \cdot \int_{0}^{1} \frac{1}{2 ( u ^{2} + 1 )} d u

Entferne die Konstante:

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

= 32 \cdot \frac{1}{2} \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)

= 32 \cdot \frac{1}{2} [arctan (u)]_{0}^{1}

Vereinfache

32 \cdot \frac{1}{2} [arctan (u)]_{0}^{1} : 16 [arctan (u)]_{0}^{1}

= 16 [arctan (u)]_{0}^{1}

Berechne die Grenzen:

\frac{π}{4}

= 16 \cdot \frac{π}{4}

Vereinfache

= 4 π

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