integral from 2 to 7 of 5log_{6}(x)

Lösung

\int_{2}^{7} 5 lo g_{6} (x) d x

\frac{5 ( 7 ln ( 7 ) - 5 - 2 ln ( 2 ) )}{ln ( 6 )}

Dezimale

20.18986 \dots

Schritte zur Lösung

\int_{2}^{7} 5 lo g_{6} (x) d x

Entferne die Konstante:

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

= 5 \cdot \int_{2}^{7} lo g_{6} (x) d x

Wende die partielle Integration an

= 5 [x lo g_{6} (x) - \int \frac{1}{ln ( 6 )} d x]_{2}^{7}

\int \frac{1}{ln ( 6 )} d x = \frac{1}{ln ( 6 )} x

= 5 [x lo g_{6} (x) - \frac{1}{ln ( 6 )} x]_{2}^{7}

Füge

x lo g_{6} (x) - \frac{1}{ln ( 6 )} x

zusammen:

\frac{- x + x ln ( x )}{ln ( 6 )}

= 5 [\frac{- x + x ln ( x )}{ln ( 6 )}]_{2}^{7}

Berechne die Grenzen:

\frac{7 ln ( 7 ) - 5 - 2 ln ( 2 )}{ln ( 6 )}

= 5 \cdot \frac{7 ln ( 7 ) - 5 - 2 ln ( 2 )}{ln ( 6 )}

Vereinfache

= \frac{5 ( 7 ln ( 7 ) - 5 - 2 ln ( 2 ) )}{ln ( 6 )}

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