与えられた8つの不定積分を計算します。 (1) $\int (3x^3 - 2x^2) dx$ (2) $\int (x+2)^2 dx$ (3) $\int \frac{x+2}{x^3} dx$ (4) $\int \frac{x+2}{\sqrt{x}} dx$ (5) $\int x(\sqrt{x}+1)^2 dx$ (6) $\int x(2\sqrt{x}-3)^2 dx$ (7) $\int (\sqrt{x} + \frac{1}{x})^2 dx$ (8) $\int \sqrt{x}(x+1)^2 dx$

1. 問題の内容

与えられた8つの不定積分を計算します。

(1) $\int (3x^3 - 2x^2) dx$

(2) $\int (x+2)^2 dx$

(3) $\int \frac{x+2}{x^3} dx$

(4) $\int \frac{x+2}{\sqrt{x}} dx$

(5) $\int x(\sqrt{x}+1)^2 dx$

(6) $\int x(2\sqrt{x}-3)^2 dx$

(7) $\int (\sqrt{x} + \frac{1}{x})^2 dx$

(8) $\int \sqrt{x}(x+1)^2 dx$

2. 解き方の手順

(1) $\int (3x^3 - 2x^2) dx$

積分を計算します。

$\int 3x^3 dx - \int 2x^2 dx = 3\int x^3 dx - 2\int x^2 dx = 3\cdot \frac{x^4}{4} - 2\cdot \frac{x^3}{3} + C$

(2) $\int (x+2)^2 dx$

$(x+2)^2 = x^2 + 4x + 4$を展開します。

$\int (x^2 + 4x + 4) dx = \int x^2 dx + 4\int x dx + 4\int dx = \frac{x^3}{3} + 4\cdot \frac{x^2}{2} + 4x + C = \frac{x^3}{3} + 2x^2 + 4x + C$

(3) $\int \frac{x+2}{x^3} dx$

$\frac{x+2}{x^3} = \frac{x}{x^3} + \frac{2}{x^3} = \frac{1}{x^2} + \frac{2}{x^3} = x^{-2} + 2x^{-3}$

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

(4) $\int \frac{x+2}{\sqrt{x}} dx$

$\frac{x+2}{\sqrt{x}} = \frac{x}{\sqrt{x}} + \frac{2}{\sqrt{x}} = \sqrt{x} + \frac{2}{\sqrt{x}} = x^{\frac{1}{2}} + 2x^{-\frac{1}{2}}$

$\int (x^{\frac{1}{2}} + 2x^{-\frac{1}{2}}) dx = \int x^{\frac{1}{2}} dx + 2\int x^{-\frac{1}{2}} dx = \frac{x^{\frac{3}{2}}}{\frac{3}{2}} + 2\cdot \frac{x^{\frac{1}{2}}}{\frac{1}{2}} + C = \frac{2}{3}x^{\frac{3}{2}} + 4x^{\frac{1}{2}} + C$

(5) $\int x(\sqrt{x}+1)^2 dx$

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

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

$\int (x^2 + 2x^{\frac{3}{2}} + x) dx = \int x^2 dx + 2\int x^{\frac{3}{2}} dx + \int x dx = \frac{x^3}{3} + 2\cdot \frac{x^{\frac{5}{2}}}{\frac{5}{2}} + \frac{x^2}{2} + C = \frac{x^3}{3} + \frac{4}{5}x^{\frac{5}{2}} + \frac{x^2}{2} + C$

(6) $\int x(2\sqrt{x}-3)^2 dx$

$(2\sqrt{x}-3)^2 = (2\sqrt{x})^2 - 2\cdot 2\sqrt{x} \cdot 3 + 3^2 = 4x - 12\sqrt{x} + 9$

$x(2\sqrt{x}-3)^2 = x(4x - 12\sqrt{x} + 9) = 4x^2 - 12x\sqrt{x} + 9x = 4x^2 - 12x^{\frac{3}{2}} + 9x$

$\int (4x^2 - 12x^{\frac{3}{2}} + 9x) dx = 4\int x^2 dx - 12\int x^{\frac{3}{2}} dx + 9\int x dx = 4\cdot \frac{x^3}{3} - 12\cdot \frac{x^{\frac{5}{2}}}{\frac{5}{2}} + 9\cdot \frac{x^2}{2} + C = \frac{4}{3}x^3 - \frac{24}{5}x^{\frac{5}{2}} + \frac{9}{2}x^2 + C$

(7) $\int (\sqrt{x} + \frac{1}{x})^2 dx$

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

$\int (x + 2x^{-\frac{1}{2}} + x^{-2}) dx = \int x dx + 2\int x^{-\frac{1}{2}} dx + \int x^{-2} dx = \frac{x^2}{2} + 2\cdot \frac{x^{\frac{1}{2}}}{\frac{1}{2}} + \frac{x^{-1}}{-1} + C = \frac{x^2}{2} + 4\sqrt{x} - \frac{1}{x} + C$

(8) $\int \sqrt{x}(x+1)^2 dx$

$(x+1)^2 = x^2 + 2x + 1$

$\sqrt{x}(x+1)^2 = \sqrt{x}(x^2 + 2x + 1) = x^{\frac{1}{2}}(x^2 + 2x + 1) = x^{\frac{5}{2}} + 2x^{\frac{3}{2}} + x^{\frac{1}{2}}$

$\int (x^{\frac{5}{2}} + 2x^{\frac{3}{2}} + x^{\frac{1}{2}}) dx = \int x^{\frac{5}{2}} dx + 2\int x^{\frac{3}{2}} dx + \int x^{\frac{1}{2}} dx = \frac{x^{\frac{7}{2}}}{\frac{7}{2}} + 2\cdot \frac{x^{\frac{5}{2}}}{\frac{5}{2}} + \frac{x^{\frac{3}{2}}}{\frac{3}{2}} + C = \frac{2}{7}x^{\frac{7}{2}} + \frac{4}{5}x^{\frac{5}{2}} + \frac{2}{3}x^{\frac{3}{2}} + C$

3. 最終的な答え

(1) $\frac{3}{4}x^4 - \frac{2}{3}x^3 + C$

(2) $\frac{1}{3}x^3 + 2x^2 + 4x + C$

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

(4) $\frac{2}{3}x^{\frac{3}{2}} + 4x^{\frac{1}{2}} + C$

(5) $\frac{1}{3}x^3 + \frac{4}{5}x^{\frac{5}{2}} + \frac{1}{2}x^2 + C$

(6) $\frac{4}{3}x^3 - \frac{24}{5}x^{\frac{5}{2}} + \frac{9}{2}x^2 + C$

(7) $\frac{1}{2}x^2 + 4\sqrt{x} - \frac{1}{x} + C$

(8) $\frac{2}{7}x^{\frac{7}{2}} + \frac{4}{5}x^{\frac{5}{2}} + \frac{2}{3}x^{\frac{3}{2}} + C$