integral from 1 to e of (24+ln(x))^2

解

\int_{1}^{e} (24 + ln (x))^{2} d x

577 e - 530

十進法表記

1038.44861 \dots

解答ステップ

\int_{1}^{e} (24 + ln (x))^{2} d x

24 + ln (x) = 24 (1 + \frac{ln ( x )}{24})

= \int_{1}^{e} (24 (1 + \frac{ln ( x )}{24}))^{2} d x

簡素化

(24 (1 + \frac{ln ( x )}{24}))^{2} : 576 + 48 ln (x) + ln^{2} (x)

= \int_{1}^{e} 576 + 48 ln (x) + ln^{2} (x) d x

総和規則を適用する:

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

= \int_{1}^{e} 576 d x + \int_{1}^{e} 48 ln (x) d x + \int_{1}^{e} ln^{2} (x) d x

\int_{1}^{e} 576 d x = 576 e - 576

\int_{1}^{e} 48 ln (x) d x = 48

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

= 576 e - 576 + 48 + e - 2

簡素化

= 577 e - 530