問題12.1では、次の不定積分を求めます。 (1) $y = \sqrt[3]{x}$ (2) $y = \frac{1}{x^3} + \sqrt{x}$ 問題12.2では、次の定積分の値を求めます。 (1) $I = \int_{1}^{3} (\sqrt{x} - 1) dx$ (2) $I = \int_{1}^{2} (\sqrt[3]{x^2} - \frac{1}{x^2}) dx$

1. 問題の内容

問題12.1では、次の不定積分を求めます。

(1) $y = \sqrt[3]{x}$

(2) $y = \frac{1}{x^3} + \sqrt{x}$

問題12.2では、次の定積分の値を求めます。

(1) $I = \int_{1}^{3} (\sqrt{x} - 1) dx$

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

2. 解き方の手順

問題12.1

(1) $y = \sqrt[3]{x} = x^{\frac{1}{3}}$

不定積分は

$\int x^{\frac{1}{3}} dx = \frac{x^{\frac{1}{3}+1}}{\frac{1}{3}+1} + C = \frac{x^{\frac{4}{3}}}{\frac{4}{3}} + C = \frac{3}{4}x^{\frac{4}{3}} + C$

(2) $y = \frac{1}{x^3} + \sqrt{x} = x^{-3} + x^{\frac{1}{2}}$

不定積分は

$\int (x^{-3} + x^{\frac{1}{2}}) dx = \frac{x^{-3+1}}{-3+1} + \frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1} + C = \frac{x^{-2}}{-2} + \frac{x^{\frac{3}{2}}}{\frac{3}{2}} + C = -\frac{1}{2x^2} + \frac{2}{3}x^{\frac{3}{2}} + C$

問題12.2

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

$\int (x^{\frac{1}{2}} - 1) dx = \frac{x^{\frac{3}{2}}}{\frac{3}{2}} - x + C = \frac{2}{3}x^{\frac{3}{2}} - x + C$

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

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

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

$I = [\frac{3}{5}x^{\frac{5}{3}} + \frac{1}{x}]_{1}^{2} = (\frac{3}{5}(2)^{\frac{5}{3}} + \frac{1}{2}) - (\frac{3}{5}(1)^{\frac{5}{3}} + 1) = \frac{3}{5}2\sqrt[3]{2^2} + \frac{1}{2} - \frac{3}{5} - 1 = \frac{6}{5}\sqrt[3]{4} + \frac{1}{2} - \frac{3}{5} - 1 = \frac{6}{5}\sqrt[3]{4} - \frac{1}{2} - \frac{3}{5} = \frac{6}{5}\sqrt[3]{4} - \frac{5}{10} - \frac{6}{10} = \frac{6}{5}\sqrt[3]{4} - \frac{11}{10}$

3. 最終的な答え

問題12.1

(1) $\frac{3}{4}x^{\frac{4}{3}} + C$

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

問題12.2

(1) $2\sqrt{3} - \frac{8}{3}$

(2) $\frac{6}{5}\sqrt[3]{4} - \frac{11}{10}$