与えられた複数の関数を積分せよという問題です。具体的には以下の12個の関数について不定積分を求める必要があります。 (1) $(x-3)(2x-1)$ (2) $(\frac{x+1}{x})^2$ (3) $\frac{(x-1)(\sqrt{x}-2)}{x^2}$ (4) $(e^x+e^{-x})^2$ (5) $2^{x+1}+3^{x+1}$ (6) $\frac{1}{\cos^2 x \sin^2 x}$ (7) $\cos^2 \frac{x}{2}$ (8) $\sin^3 x$ (9) $\sin x \cos 2x$ (10) $\frac{1}{\tan^2 x}$ (11) $\frac{x + \sqrt{x^2-1}}{x - \sqrt{x^2-1}}$ (12) $\frac{\sqrt{1+x^2} + \sqrt{1-x^2}}{\sqrt{1-x^4}}$

はい、承知いたしました。問題の指示に従い、画像の数学の問題を解いていきます。

1. 問題の内容

与えられた複数の関数を積分せよという問題です。具体的には以下の12個の関数について不定積分を求める必要があります。

(1) $(x-3)(2x-1)$

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

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

(4) $(e^x+e^{-x})^2$

(5) $2^{x+1}+3^{x+1}$

(6) $\frac{1}{\cos^2 x \sin^2 x}$

(7) $\cos^2 \frac{x}{2}$

(8) $\sin^3 x$

(9) $\sin x \cos 2x$

(10) $\frac{1}{\tan^2 x}$

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

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

2. 解き方の手順

それぞれの関数について、積分を計算します。

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

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

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

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

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

(6) $\int \frac{1}{\cos^2 x \sin^2 x} dx = \int \frac{\sin^2 x + \cos^2 x}{\cos^2 x \sin^2 x} dx = \int (\frac{1}{\cos^2 x} + \frac{1}{\sin^2 x}) dx = \tan x - \cot x + C$

(7) $\int \cos^2 \frac{x}{2} dx = \int \frac{1 + \cos x}{2} dx = \frac{1}{2}x + \frac{1}{2}\sin x + C$

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

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

(10) $\int \frac{1}{\tan^2 x} dx = \int \cot^2 x dx = \int (\frac{1}{\sin^2 x} - 1) dx = -\cot x - x + C$

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

(12) $\int \frac{\sqrt{1+x^2} + \sqrt{1-x^2}}{\sqrt{1-x^4}} dx = \int (\frac{\sqrt{1+x^2}}{\sqrt{1-x^2}\sqrt{1+x^2}} + \frac{\sqrt{1-x^2}}{\sqrt{1-x^2}\sqrt{1+x^2}}) dx = \int (\frac{1}{\sqrt{1-x^2}} + \frac{1}{\sqrt{1+x^2}}) dx = \arcsin x + \sinh^{-1} x + C = \arcsin x + \ln(x + \sqrt{1+x^2}) + C$

3. 最終的な答え

(1) $\frac{2}{3}x^3 - \frac{7}{2}x^2 + 3x + C$

(2) $x + 2\ln|x| - \frac{1}{x} + C$

(3) $2\sqrt{x} - 2\ln|x| + \frac{2}{\sqrt{x}} - \frac{2}{x} + C$

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

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

(6) $\tan x - \cot x + C$

(7) $\frac{1}{2}x + \frac{1}{2}\sin x + C$

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

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

(10) $-\cot x - x + C$

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

(12) $\arcsin x + \ln(x + \sqrt{1+x^2}) + C$