(1) $\sin(-\frac{5\pi}{6})$ (2) $\cos(\frac{3\pi}{4})$ (3) $\tan(\frac{7\pi}{6})$

はい、承知いたしました。画像に写っている数学の問題を解いていきます。

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1. 問題の内容**

与えられた三角関数の値を求め、対数の計算をし、関数を微分し、そして不定積分を計算する問題です。具体的には、以下の問題を解きます。

1. 三角関数の値:

(1) $\sin(-\frac{5\pi}{6})$

(2) $\cos(\frac{3\pi}{4})$

(3) $\tan(\frac{7\pi}{6})$

2. 対数の計算:

(1) $\log_{2/7} 4 + 3\log_{2/7} 2$

(2) $2\log_{10} \frac{\sqrt{3}}{10} - \log_{10} 30$

3. 関数の微分:

(1) $y = x^4$

(2) $y = 3\sqrt{x}$

(3) $y = \cos x$

(4) $y = x^3 \sin x$

(5) $y = \tan x + e^x$

(6) $y = \cos x^2$

(7) $y = (x^2 + 1)^{-1/3}$

(8) $y = x^x$

(9) $y = xe^{2x+3}$

4. 不定積分:

(1) $\int (x^2 + 1) \, dx$

(2) $\int \cos x \, dx$

(3) $\int xe^x \, dx$

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

(5) $\int \frac{1}{x \log x} \, dx$

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

(7) $\int_{-1}^{2} 4x \, dx$

(8) $\int_{0}^{\pi} \sin x \, dx$

(9) $\int_{1}^{e} \frac{(\log x)^2}{x} \, dx$

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2. 解き方の手順**

1. **三角関数の値**

(1) $\sin(-\frac{5\pi}{6})$:

$\sin(-\frac{5\pi}{6}) = -\sin(\frac{5\pi}{6}) = -\sin(\pi - \frac{\pi}{6}) = -\sin(\frac{\pi}{6}) = -\frac{1}{2}$

(2) $\cos(\frac{3\pi}{4})$:

$\cos(\frac{3\pi}{4}) = \cos(\pi - \frac{\pi}{4}) = -\cos(\frac{\pi}{4}) = -\frac{\sqrt{2}}{2}$

(3) $\tan(\frac{7\pi}{6})$:

$\tan(\frac{7\pi}{6}) = \tan(\pi + \frac{\pi}{6}) = \tan(\frac{\pi}{6}) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$

2. **対数の計算**

(1) $\log_{2/7} 4 + 3\log_{2/7} 2$:

$\log_{2/7} 4 + 3\log_{2/7} 2 = \log_{2/7} 4 + \log_{2/7} 2^3 = \log_{2/7} 4 + \log_{2/7} 8 = \log_{2/7} (4 \times 8) = \log_{2/7} 32 = \log_{2/7} (\frac{2}{7})^{-5} = -5$

(2) $2\log_{10} \frac{\sqrt{3}}{10} - \log_{10} 30$:

$2\log_{10} \frac{\sqrt{3}}{10} - \log_{10} 30 = \log_{10} (\frac{\sqrt{3}}{10})^2 - \log_{10} 30 = \log_{10} \frac{3}{100} - \log_{10} 30 = \log_{10} \frac{3/100}{30} = \log_{10} \frac{1}{1000} = \log_{10} 10^{-3} = -3$

3. **関数の微分**

(1) $y = x^4$:

$y' = 4x^3$

(2) $y = 3\sqrt{x} = 3x^{1/2}$:

$y' = 3 \times \frac{1}{2} x^{-1/2} = \frac{3}{2\sqrt{x}}$

(3) $y = \cos x$:

$y' = -\sin x$

(4) $y = x^3 \sin x$:

$y' = 3x^2 \sin x + x^3 \cos x$

(5) $y = \tan x + e^x$:

$y' = \sec^2 x + e^x = \frac{1}{\cos^2 x} + e^x$

(6) $y = \cos(x^2)$:

$y' = -\sin(x^2) \times 2x = -2x \sin(x^2)$

(7) $y = (x^2 + 1)^{-1/3}$:

$y' = -\frac{1}{3} (x^2 + 1)^{-4/3} \times 2x = -\frac{2x}{3(x^2 + 1)^{4/3}}$

(8) $y = x^x$:

$\log y = x \log x$

$\frac{y'}{y} = \log x + x \times \frac{1}{x} = \log x + 1$

$y' = y(\log x + 1) = x^x(\log x + 1)$

(9) $y = xe^{2x+3}$:

$y' = e^{2x+3} + x e^{2x+3} \times 2 = e^{2x+3}(1 + 2x)$

4. **不定積分**

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

(2) $\int \cos x \, dx = \sin x + C$

(3) $\int xe^x \, dx$: (部分積分)

$u = x, dv = e^x dx$, $du = dx, v = e^x$

$\int xe^x \, dx = xe^x - \int e^x \, dx = xe^x - e^x + C = (x-1)e^x + C$

(4) $\int 2\sqrt{x} \, dx = 2 \int x^{1/2} \, dx = 2 \times \frac{2}{3} x^{3/2} + C = \frac{4}{3} x^{3/2} + C = \frac{4}{3} \sqrt{x^3} + C$

(5) $\int \frac{1}{x \log x} \, dx$: (置換積分)

$u = \log x$, $du = \frac{1}{x} dx$

$\int \frac{1}{x \log x} \, dx = \int \frac{1}{u} \, du = \log |u| + C = \log |\log x| + C$

(6) $\int \sqrt{2x - 3} \, dx$: (置換積分)

$u = 2x - 3$, $du = 2 dx$, $dx = \frac{1}{2} du$

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

(7) $\int_{-1}^{2} 4x \, dx = [2x^2]_{-1}^{2} = 2(2^2) - 2(-1)^2 = 2(4) - 2(1) = 8 - 2 = 6$

(8) $\int_{0}^{\pi} \sin x \, dx = [-\cos x]_{0}^{\pi} = -\cos(\pi) - (-\cos(0)) = -(-1) - (-1) = 1 + 1 = 2$

(9) $\int_{1}^{e} \frac{(\log x)^2}{x} \, dx$: (置換積分)

$u = \log x$, $du = \frac{1}{x} dx$

$x = 1 \Rightarrow u = \log 1 = 0$

$x = e \Rightarrow u = \log e = 1$

$\int_{1}^{e} \frac{(\log x)^2}{x} \, dx = \int_{0}^{1} u^2 \, du = [\frac{u^3}{3}]_{0}^{1} = \frac{1^3}{3} - \frac{0^3}{3} = \frac{1}{3}$

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3. 最終的な答え**

1. 三角関数の値:

(1) $-\frac{1}{2}$

(2) $-\frac{\sqrt{2}}{2}$

(3) $\frac{\sqrt{3}}{3}$

2. 対数の計算:

(1) $-5$

(2) $-3$

3. 関数の微分:

(1) $4x^3$

(2) $\frac{3}{2\sqrt{x}}$

(3) $-\sin x$

(4) $3x^2 \sin x + x^3 \cos x$

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

(6) $-2x \sin(x^2)$

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

(8) $x^x(\log x + 1)$

(9) $e^{2x+3}(1 + 2x)$

4. 不定積分:

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

(2) $\sin x + C$

(3) $(x-1)e^x + C$

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

(5) $\log |\log x| + C$

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

(7) 6

(8) 2

(9) $\frac{1}{3}$