integral from 0 to 1 of xpi^x

Lösung

\int_{0}^{1} x π^{x} d x

\frac{π}{ln ( π )} + \frac{- π + 1}{ln ^{2} ( π )}

Dezimale

1.11010 \dots

Schritte zur Lösung

\int_{0}^{1} x π^{x} d x

Wende die partielle Integration an

= [\frac{π ^{x} x}{ln ( π )} - \int \frac{π ^{x}}{ln ( π )} d x]_{0}^{1}

\int \frac{π ^{x}}{ln ( π )} d x = \frac{1}{ln ^{2} ( π )} π^{x}

= [\frac{π ^{x} x}{ln ( π )} - \frac{1}{ln ^{2} ( π )} π^{x}]_{0}^{1}

Berechne die Grenzen:

\frac{π}{ln ( π )} - \frac{π}{ln ^{2} ( π )} + \frac{1}{ln ^{2} ( π )}

= \frac{π}{ln ( π )} - \frac{π}{ln ^{2} ( π )} + \frac{1}{ln ^{2} ( π )}

Vereinfache

= \frac{π}{ln ( π )} + \frac{- π + 1}{ln ^{2} ( π )}

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