与えられた5つの不定積分を計算し、空欄を埋める問題です。

1. 問題の内容

与えられた5つの不定積分を計算し、空欄を埋める問題です。

2. 解き方の手順

(1) $\int \frac{(x+2)^3}{x^3} dx = \int \frac{x^3+6x^2+12x+8}{x^3} dx = \int (1+\frac{6}{x}+\frac{12}{x^2}+\frac{8}{x^3}) dx$

$= \int (1+\frac{6}{x}+12x^{-2}+8x^{-3}) dx = x+6\log|x|+12\frac{x^{-1}}{-1}+8\frac{x^{-2}}{-2}+C$

$= x+6\log|x|-\frac{12}{x}-\frac{4}{x^2}+C$

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

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

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

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

$= \frac{5}{2}xe^{2x} + \frac{3}{2}e^{2x} - \frac{5}{4}e^{2x}+C = \frac{5}{2}xe^{2x} + \frac{1}{4}e^{2x}+C = (\frac{10}{4}x+\frac{1}{4})e^{2x}+C$

(4) $\int \log(3x) dx = \int (\log 3 + \log x) dx = (\log 3)x + \int \log x dx$

$\int \log x dx = x\log x - \int x\frac{1}{x} dx = x\log x - \int 1 dx = x\log x - x + C$

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

$\int \log(3x) dx = x\log(3x) - x + C = 1\cdot x \log(3x) - 1\cdot x + C$

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

$= 1 \cdot (x\log(3x)-x)+C = 1((1x-0)\log(3x) - x) + C$.

Then $x(\log(3x) - 1)+C$.

$I = (x(\log(3x) - 1)+C$. $\frac{d}{dx} I = \log(3x)-1 +x \frac{3}{3x} = \log(3x)-1+1 = \log(3x)$.

(5) $\int \sin 6x \cos 2x dx = \int \frac{1}{2}(\sin(6x+2x)+\sin(6x-2x)) dx = \frac{1}{2}\int (\sin 8x + \sin 4x) dx$

$= \frac{1}{2}(-\frac{1}{8}\cos 8x - \frac{1}{4}\cos 4x) + C = -\frac{1}{16}\cos 8x - \frac{1}{8}\cos 4x + C$

3. 最終的な答え

(1) 1:6, 2:12, 3:1, 4:4

(2) 5:3

(3) 6:10, 7:4, 8:1, 9:2

(4) 10:1, 11:1, 12:0, 13:3, 14:0

(5) 15:1, 16:16, 17:8, 18:1, 19:8