integral from 0 to 1 of-ln(1-x)

Lösung

\int_{0}^{1} - ln (1 - x) d x

1

Schritte zur Lösung

\int_{0}^{1} - ln (1 - x) d x

Entferne die Konstante:

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

= - \int_{0}^{1} ln (1 - x) d x

Wende U-Substitution an

= - \int_{1}^{0} - ln (u) d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - (- \int_{0}^{1} - ln (u) d u)

Entferne die Konstante:

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

= - (- (- \int_{0}^{1} ln (u) d u))

Wende die partielle Integration an

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

\int 1 d u = u

= - (- (- [u ln (u) - u]_{0}^{1}))

Vereinfache

= - [u ln (u) - u]_{0}^{1}

Berechne die Grenzen:

- 1

= - (- 1)

Vereinfache

= 1

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