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

解

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

- arctan (22) + 22

十進法表記

1.59746 \dots

解答ステップ

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

u置換積分法を適用する

= \int_{0}^{22} \frac{u ^{2}}{u ^{2} + 1} d u

\frac{u ^{2}}{u ^{2} + 1} = - \frac{1}{u ^{2} + 1} + 1

= \int_{0}^{22} - \frac{1}{u ^{2} + 1} + 1 d u

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_{0}^{22} \frac{1}{u ^{2} + 1} d u + \int_{0}^{22} 1 d u

\int_{0}^{22} \frac{1}{u ^{2} + 1} d u = arctan (22)

\int_{0}^{22} 1 d u = 22

= - arctan (22) + 22