integral from-5 to 5 of 2/(x^2+2)

Lösung

\int_{- 5}^{5} \frac{2}{x ^{2} + 2} d x

22 arctan (\frac{5}{2})

Dezimale

3.66324 \dots

Schritte zur Lösung

\int_{- 5}^{5} \frac{2}{x ^{2} + 2} d x

Entferne die Konstante:

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

= 2 \cdot \int_{- 5}^{5} \frac{1}{x ^{2} + 2} d x

Wende integrale Substitution an

= 2 \cdot \int_{- \frac{5}{2}}^{\frac{5}{2}} \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

= 2 \cdot \frac{1}{2} \cdot \int_{- \frac{5}{2}}^{\frac{5}{2}} \frac{1}{u ^{2} + 1} d u

Nutze das gemeinsame Integral :

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

= 2 \cdot \frac{1}{2} [arctan (u)]_{- \frac{5}{2}}^{\frac{5}{2}}

Vereinfache

2 \cdot \frac{1}{2} [arctan (u)]_{- \frac{5}{2}}^{\frac{5}{2}} : 2 [arctan (u)]_{- \frac{5}{2}}^{\frac{5}{2}}

= 2 [arctan (u)]_{- \frac{5}{2}}^{\frac{5}{2}}

Berechne die Grenzen:

2 arctan (\frac{5}{2})

= 2 \cdot 2 arctan (\frac{5}{2})

Vereinfache

= 22 arctan (\frac{5}{2})

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