integral from 0 to 4 of xsqrt(x+1)

解

\int_{0}^{4} x x + 1 d x

105 - \frac{2}{5} - \frac{10 5 - 2}{3}

十進法表記

15.17378 \dots

解答ステップ

\int_{0}^{4} x x + 1 d x

u置換積分法を適用する

= \int_{1}^{5} (u - 1) u d u

拡張

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

= \int_{1}^{5} u^{\frac{3}{2}} - u 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_{1}^{5} u^{\frac{3}{2}} d u - \int_{1}^{5} u d u

\int_{1}^{5} u^{\frac{3}{2}} d u = 105 - \frac{2}{5}

\int_{1}^{5} u d u = \frac{10 5 - 2}{3}

= 105 - \frac{2}{5} - \frac{10 5 - 2}{3}