以下の6つの定積分を計算する問題です。 (1) $\int_{-1}^{2} x^2 x^3 dx$ (2) $\int_{1}^{2} \frac{1}{x^4} dx$ (3) $\int_{1}^{4} \frac{1}{\sqrt{x^3}} dx$ (4) $\int_{0}^{1} x \sqrt{x} dx$ (5) $\int_{-2}^{2} \frac{1}{(x-3)^2} dx$ (6) $\int_{-1}^{1} \sqrt[4]{x+2} dx$

1. 問題の内容

以下の6つの定積分を計算する問題です。

(1) $\int_{-1}^{2} x^2 x^3 dx$

(2) $\int_{1}^{2} \frac{1}{x^4} dx$

(3) $\int_{1}^{4} \frac{1}{\sqrt{x^3}} dx$

(4) $\int_{0}^{1} x \sqrt{x} dx$

(5) $\int_{-2}^{2} \frac{1}{(x-3)^2} dx$

(6) $\int_{-1}^{1} \sqrt[4]{x+2} dx$

2. 解き方の手順

(1) $\int_{-1}^{2} x^2 x^3 dx = \int_{-1}^{2} x^5 dx$

$= [\frac{x^6}{6}]_{-1}^{2} = \frac{2^6}{6} - \frac{(-1)^6}{6} = \frac{64}{6} - \frac{1}{6} = \frac{63}{6} = \frac{21}{2}$

(2) $\int_{1}^{2} \frac{1}{x^4} dx = \int_{1}^{2} x^{-4} dx$

$= [\frac{x^{-3}}{-3}]_{1}^{2} = [\frac{-1}{3x^3}]_{1}^{2} = \frac{-1}{3(2^3)} - \frac{-1}{3(1^3)} = \frac{-1}{24} + \frac{1}{3} = \frac{-1+8}{24} = \frac{7}{24}$

(3) $\int_{1}^{4} \frac{1}{\sqrt{x^3}} dx = \int_{1}^{4} x^{-\frac{3}{2}} dx$

$= [\frac{x^{-\frac{1}{2}}}{-\frac{1}{2}}]_{1}^{4} = [-2x^{-\frac{1}{2}}]_{1}^{4} = [\frac{-2}{\sqrt{x}}]_{1}^{4} = \frac{-2}{\sqrt{4}} - \frac{-2}{\sqrt{1}} = \frac{-2}{2} + 2 = -1 + 2 = 1$

(4) $\int_{0}^{1} x \sqrt{x} dx = \int_{0}^{1} x^{\frac{3}{2}} dx$

$= [\frac{x^{\frac{5}{2}}}{\frac{5}{2}}]_{0}^{1} = [\frac{2}{5} x^{\frac{5}{2}}]_{0}^{1} = \frac{2}{5}(1^{\frac{5}{2}}) - \frac{2}{5}(0^{\frac{5}{2}}) = \frac{2}{5} - 0 = \frac{2}{5}$

(5) $\int_{-2}^{2} \frac{1}{(x-3)^2} dx = \int_{-2}^{2} (x-3)^{-2} dx$

$= [\frac{(x-3)^{-1}}{-1}]_{-2}^{2} = [\frac{-1}{x-3}]_{-2}^{2} = \frac{-1}{2-3} - \frac{-1}{-2-3} = \frac{-1}{-1} - \frac{-1}{-5} = 1 - \frac{1}{5} = \frac{4}{5}$

(6) $\int_{-1}^{1} \sqrt[4]{x+2} dx = \int_{-1}^{1} (x+2)^{\frac{1}{4}} dx$

$= [\frac{(x+2)^{\frac{5}{4}}}{\frac{5}{4}}]_{-1}^{1} = [\frac{4}{5} (x+2)^{\frac{5}{4}}]_{-1}^{1} = \frac{4}{5}(1+2)^{\frac{5}{4}} - \frac{4}{5}(-1+2)^{\frac{5}{4}} = \frac{4}{5}(3^{\frac{5}{4}}) - \frac{4}{5}(1^{\frac{5}{4}}) = \frac{4}{5}(3^{\frac{5}{4}}) - \frac{4}{5} = \frac{4}{5}(3^{\frac{5}{4}}-1)$

3. 最終的な答え

(1) $\frac{21}{2}$

(2) $\frac{7}{24}$

(3) $1$

(4) $\frac{2}{5}$

(5) $\frac{4}{5}$

(6) $\frac{4}{5}(3^{\frac{5}{4}}-1)$