与えられた6つの不定積分を求めよ。 (1) $\int \frac{x}{(x-3)^2} dx$ (2) $\int x\sqrt{x-2} dx$ (3) $\int (3x+2)\sqrt{x+1} dx$ (4) $\int (x-2)\sqrt{3-2x} dx$ (5) $\int \frac{x-2}{\sqrt{x+1}} dx$ (6) $\int x \sqrt[3]{x+2} dx$

了解しました。画像の問題を解きます。

1. 問題の内容

与えられた6つの不定積分を求めよ。

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

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

(3) $\int (3x+2)\sqrt{x+1} dx$

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

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

(6) $\int x \sqrt[3]{x+2} dx$

2. 解き方の手順

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

$t = x - 3$ と置換すると、$x = t + 3$，$dx = dt$。

$\int \frac{t+3}{t^2} dt = \int (\frac{1}{t} + \frac{3}{t^2}) dt = \int (\frac{1}{t} + 3t^{-2}) dt = \ln|t| - 3t^{-1} + C = \ln|x-3| - \frac{3}{x-3} + C$

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

$t = x - 2$ と置換すると、$x = t + 2$，$dx = dt$。

$\int (t+2)\sqrt{t} dt = \int (t^{3/2} + 2t^{1/2}) dt = \frac{2}{5}t^{5/2} + \frac{4}{3}t^{3/2} + C = \frac{2}{5}(x-2)^{5/2} + \frac{4}{3}(x-2)^{3/2} + C$

(3) $\int (3x+2)\sqrt{x+1} dx$

$t = x + 1$ と置換すると、$x = t - 1$，$dx = dt$。

$\int (3(t-1)+2)\sqrt{t} dt = \int (3t-1)\sqrt{t} dt = \int (3t^{3/2} - t^{1/2}) dt = \frac{6}{5}t^{5/2} - \frac{2}{3}t^{3/2} + C = \frac{6}{5}(x+1)^{5/2} - \frac{2}{3}(x+1)^{3/2} + C$

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

$t = 3 - 2x$ と置換すると、$x = \frac{3-t}{2}$，$dx = -\frac{1}{2} dt$。

$\int (\frac{3-t}{2} - 2)\sqrt{t} (-\frac{1}{2}) dt = \int (\frac{3-t-4}{2})(-\frac{1}{2})\sqrt{t} dt = \int (\frac{-t-1}{2})(-\frac{1}{2})\sqrt{t} dt = \frac{1}{4}\int (t^{3/2} + t^{1/2}) dt = \frac{1}{4}(\frac{2}{5}t^{5/2} + \frac{2}{3}t^{3/2}) + C = \frac{1}{10}(3-2x)^{5/2} + \frac{1}{6}(3-2x)^{3/2} + C$

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

$t = x + 1$ と置換すると、$x = t - 1$，$dx = dt$。

$\int \frac{t-1-2}{\sqrt{t}} dt = \int \frac{t-3}{\sqrt{t}} dt = \int (t^{1/2} - 3t^{-1/2}) dt = \frac{2}{3}t^{3/2} - 6t^{1/2} + C = \frac{2}{3}(x+1)^{3/2} - 6(x+1)^{1/2} + C$

(6) $\int x \sqrt[3]{x+2} dx$

$t = x + 2$ と置換すると、$x = t - 2$，$dx = dt$。

$\int (t-2)\sqrt[3]{t} dt = \int (t^{4/3} - 2t^{1/3}) dt = \frac{3}{7}t^{7/3} - \frac{3}{2}t^{4/3} + C = \frac{3}{7}(x+2)^{7/3} - \frac{3}{2}(x+2)^{4/3} + C$

3. 最終的な答え

(1) $\ln|x-3| - \frac{3}{x-3} + C$

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

(3) $\frac{6}{5}(x+1)^{5/2} - \frac{2}{3}(x+1)^{3/2} + C$

(4) $\frac{1}{10}(3-2x)^{5/2} + \frac{1}{6}(3-2x)^{3/2} + C$

(5) $\frac{2}{3}(x+1)^{3/2} - 6(x+1)^{1/2} + C$

(6) $\frac{3}{7}(x+2)^{7/3} - \frac{3}{2}(x+2)^{4/3} + C$