integral from 3 to 7 of-2log_{10}(x)

Lösung

\int_{3}^{7} - 2 lo g_{10} (x) d x

- \frac{2 ( 7 ln ( 7 ) - 4 - 3 ln ( 3 ) )}{ln ( 10 )}

Dezimale

- 5.49428 \dots

Schritte zur Lösung

\int_{3}^{7} - 2 lo g_{10} (x) d x

Entferne die Konstante:

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

= - 2 \cdot \int_{3}^{7} lo g_{10} (x) d x

Wende die partielle Integration an

= - 2 [x lo g_{10} (x) - \int \frac{1}{ln ( 10 )} d x]_{3}^{7}

\int \frac{1}{ln ( 10 )} d x = \frac{1}{ln ( 10 )} x

= - 2 [x lo g_{10} (x) - \frac{1}{ln ( 10 )} x]_{3}^{7}

Füge

x lo g_{10} (x) - \frac{1}{ln ( 10 )} x

zusammen:

\frac{- x + x ln ( x )}{ln ( 10 )}

= - 2 [\frac{- x + x ln ( x )}{ln ( 10 )}]_{3}^{7}

Berechne die Grenzen:

\frac{7 ln ( 7 ) - 4 - 3 ln ( 3 )}{ln ( 10 )}

= - 2 \cdot \frac{7 ln ( 7 ) - 4 - 3 ln ( 3 )}{ln ( 10 )}

Vereinfache

= - \frac{2 ( 7 ln ( 7 ) - 4 - 3 ln ( 3 ) )}{ln ( 10 )}

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