integral from 1 to 3 of ((x-1)^3)/(x^2)

解

\int_{1}^{3} \frac{( x - 1 ) ^{3}}{x ^{2}} d x

3 ln (3) - \frac{8}{3}

十進法表記

0.62917 \dots

解答ステップ

\int_{1}^{3} \frac{( x - 1 ) ^{3}}{x ^{2}} d x

拡張

\frac{( x - 1 ) ^{3}}{x ^{2}} : x - 3 + \frac{3}{x} - \frac{1}{x ^{2}}

= \int_{1}^{3} x - 3 + \frac{3}{x} - \frac{1}{x ^{2}} 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}^{3} x d x - \int_{1}^{3} 3 d x + \int_{1}^{3} \frac{3}{x} d x - \int_{1}^{3} \frac{1}{x ^{2}} d x

\int_{1}^{3} x d x = 4

\int_{1}^{3} 3 d x = 6

\int_{1}^{3} \frac{3}{x} d x = 3 ln (3)

\int_{1}^{3} \frac{1}{x ^{2}} d x = \frac{2}{3}

= 4 - 6 + 3 ln (3) - \frac{2}{3}

簡素化

4 - 6 + 3 ln (3) - \frac{2}{3} : 3 ln (3) - \frac{8}{3}

= 3 ln (3) - \frac{8}{3}