integral from 0 to 80 of (1-t/(80))^a

Lösung

\int_{0}^{80} (1 - \frac{t}{80})^{a} d t

\frac{80}{a + 1}

Schritte zur Lösung

\int_{0}^{80} (1 - \frac{t}{80})^{a} d t

Wende U-Substitution an

= \int_{0}^{1} (1 - u)^{a} \cdot 80 d u

Entferne die Konstante:

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

= 80 \cdot \int_{0}^{1} (1 - u)^{a} d u

Wende U-Substitution an

= 80 \cdot \int_{1}^{0} - v^{a} d v

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

= 80 (- \int_{0}^{1} - v^{a} d v)

Entferne die Konstante:

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

= 80 (- (- \int_{0}^{1} v^{a} d v))

Wende die Potenzregel an

= 80 (- (- [\frac{v ^{a + 1}}{a + 1}]_{0}^{1}))

Vereinfache

= 80 [\frac{v ^{a + 1}}{a + 1}]_{0}^{1}

Berechne die Grenzen:

\frac{1}{a + 1}

= 80 \cdot \frac{1}{a + 1}

Vereinfache

= \frac{80}{a + 1}

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