integral from 2 to 3 of 1/(xsqrt(x^4-4))

Lösung

\int_{2}^{3} \frac{1}{x x ^{4} - 4} d x

\frac{1}{4} (arctan (\frac{77}{2}) - \frac{π}{3})

Dezimale

0.07487 \dots

Schritte zur Lösung

\int_{2}^{3} \frac{1}{x x ^{4} - 4} d x

Wende U-Substitution an

= \int_{12}^{77} \frac{1}{4 ( u + 4 ) u} d u

Entferne die Konstante:

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

= \frac{1}{4} \cdot \int_{12}^{77} \frac{1}{( u + 4 ) u} d u

Wende U-Substitution an

= \frac{1}{4} \cdot \int_{23}^{77} \frac{2}{v ^{2} + 4} d v

Entferne die Konstante:

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

= \frac{1}{4} \cdot 2 \cdot \int_{23}^{77} \frac{1}{v ^{2} + 4} d v

Wende integrale Substitution an

= \frac{1}{4} \cdot 2 \cdot \int_{3}^{\frac{77}{2}} \frac{1}{2 ( w ^{2} + 1 )} d w

Entferne die Konstante:

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

= \frac{1}{4} \cdot 2 \cdot \frac{1}{2} \cdot \int_{3}^{\frac{77}{2}} \frac{1}{w ^{2} + 1} d w

Nutze das gemeinsame Integral :

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

= \frac{1}{4} \cdot 2 \cdot \frac{1}{2} [arctan (w)]_{3}^{\frac{77}{2}}

Vereinfache

\frac{1}{4} \cdot 2 \cdot \frac{1}{2} [arctan (w)]_{3}^{\frac{77}{2}} : \frac{1}{4} [arctan (w)]_{3}^{\frac{77}{2}}

= \frac{1}{4} [arctan (w)]_{3}^{\frac{77}{2}}

Berechne die Grenzen:

arctan (\frac{77}{2}) - \frac{π}{3}

= \frac{1}{4} (arctan (\frac{77}{2}) - \frac{π}{3})

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