与えられた6つの定積分を計算します。 (1) $\int_{-1}^2 x^2 x^3 dx$ (2) $\int_1^2 \frac{1}{x^4} dx$ (3) $\int_1^4 \frac{1}{\sqrt[3]{x}} dx$ (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

与えられた6つの定積分を計算します。

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

(2) $\int_1^2 \frac{1}{x^4} dx$

(3) $\int_1^4 \frac{1}{\sqrt[3]{x}} dx$

(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) $\int_{-1}^2 x^2 x^3 dx = \int_{-1}^2 x^5 dx = \left[ \frac{1}{6}x^6 \right]_{-1}^2 = \frac{1}{6}(2^6 - (-1)^6) = \frac{1}{6}(64-1) = \frac{63}{6} = \frac{21}{2}$

(2) $\int_1^2 \frac{1}{x^4} dx = \int_1^2 x^{-4} dx = \left[ \frac{x^{-3}}{-3} \right]_1^2 = \left[ -\frac{1}{3x^3} \right]_1^2 = -\frac{1}{3(2^3)} - \left(-\frac{1}{3(1^3)}\right) = -\frac{1}{24} + \frac{1}{3} = -\frac{1}{24} + \frac{8}{24} = \frac{7}{24}$

(3) $\int_1^4 \frac{1}{\sqrt[3]{x}} dx = \int_1^4 x^{-1/3} dx = \left[ \frac{x^{2/3}}{2/3} \right]_1^4 = \frac{3}{2} \left[ x^{2/3} \right]_1^4 = \frac{3}{2}(4^{2/3} - 1^{2/3}) = \frac{3}{2}((2^2)^{2/3} - 1) = \frac{3}{2}(2^{4/3} - 1) = \frac{3}{2}(2 \sqrt[3]{2} - 1)$

(4) $\int_0^1 x\sqrt{x} dx = \int_0^1 x^{3/2} dx = \left[ \frac{x^{5/2}}{5/2} \right]_0^1 = \frac{2}{5} \left[ x^{5/2} \right]_0^1 = \frac{2}{5}(1^{5/2} - 0^{5/2}) = \frac{2}{5}(1 - 0) = \frac{2}{5}$

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

(6) $\int_{-1}^1 \sqrt[4]{x+2} dx = \int_{-1}^1 (x+2)^{1/4} dx = \left[ \frac{(x+2)^{5/4}}{5/4} \right]_{-1}^1 = \frac{4}{5} \left[ (x+2)^{5/4} \right]_{-1}^1 = \frac{4}{5} ((1+2)^{5/4} - (-1+2)^{5/4}) = \frac{4}{5}(3^{5/4} - 1^{5/4}) = \frac{4}{5}(3 \sqrt[4]{3} - 1)$

3. 最終的な答え

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

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

(3) $\frac{3}{2}(2 \sqrt[3]{2} - 1)$

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

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

(6) $\frac{4}{5}(3 \sqrt[4]{3} - 1)$