integral from 0 to 1 of pie^{2x}

Lösung

\int_{0}^{1} π e^{2 x} d x

\frac{π ( e ^{2} - 1 )}{2}

Dezimale

10.03590 \dots

Schritte zur Lösung

\int_{0}^{1} π e^{2 x} d x

Entferne die Konstante:

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

= π \cdot \int_{0}^{1} e^{2 x} d x

Wende U-Substitution an

= π \cdot \int_{0}^{2} e^{u} \frac{1}{2} d u

Entferne die Konstante:

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

= π \frac{1}{2} \cdot \int_{0}^{2} e^{u} d u

Nutze das gemeinsame Integral :

\int e^{u} d u = e^{u}

= π \frac{1}{2} [e^{u}]_{0}^{2}

Vereinfache

π \frac{1}{2} [e^{u}]_{0}^{2} : \frac{π}{2} [e^{u}]_{0}^{2}

= \frac{π}{2} [e^{u}]_{0}^{2}

Berechne die Grenzen:

e^{2} - 1

= \frac{π}{2} (e^{2} - 1)

Vereinfache

= \frac{π ( e ^{2} - 1 )}{2}

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