integral from 0 to 1 of 10x5^x

Lösung

\int_{0}^{1} 10 x 5^{x} d x

\frac{10 ( 5 ln ( 5 ) - 4 )}{ln ^{2} ( 5 )}

Dezimale

15.62446 \dots

Schritte zur Lösung

\int_{0}^{1} 10 x \cdot 5^{x} d x

Entferne die Konstante:

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

= 10 \cdot \int_{0}^{1} x \cdot 5^{x} d x

Wende die partielle Integration an

= 10 [\frac{5 ^{x} x}{ln ( 5 )} - \int \frac{5 ^{x}}{ln ( 5 )} d x]_{0}^{1}

\int \frac{5 ^{x}}{ln ( 5 )} d x = \frac{5 ^{x}}{ln ^{2} ( 5 )}

= 10 [\frac{5 ^{x} x}{ln ( 5 )} - \frac{5 ^{x}}{ln ^{2} ( 5 )}]_{0}^{1}

Berechne die Grenzen:

\frac{5 ln ( 5 ) - 4}{ln ^{2} ( 5 )}

= 10 \cdot \frac{5 ln ( 5 ) - 4}{ln ^{2} ( 5 )}

Vereinfache

= \frac{10 ( 5 ln ( 5 ) - 4 )}{ln ^{2} ( 5 )}

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