integral from 1 to e of (15+ln(x))^2

Lösung

\int_{1}^{e} (15 + ln (x))^{2} d x

226 e - 197

Dezimale

417.33169 \dots

Schritte zur Lösung

\int_{1}^{e} (15 + ln (x))^{2} d x

15 + ln (x) = 15 (1 + \frac{ln ( x )}{15})

= \int_{1}^{e} (15 (1 + \frac{ln ( x )}{15}))^{2} d x

Vereinfache

(15 (1 + \frac{ln ( x )}{15}))^{2} : 225 + 30 ln (x) + ln^{2} (x)

= \int_{1}^{e} 225 + 30 ln (x) + ln^{2} (x) d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{1}^{e} 225 d x + \int_{1}^{e} 30 ln (x) d x + \int_{1}^{e} ln^{2} (x) d x

\int_{1}^{e} 225 d x = 225 e - 225

\int_{1}^{e} 30 ln (x) d x = 30

\int_{1}^{e} ln^{2} (x) d x = e - 2

= 225 e - 225 + 30 + e - 2

Vereinfache

= 226 e - 197

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