同济高数 第4章 第4-1-2题

[AI解答]

[AI解答]

以下为各题详细解答过程，使用基本积分公式与代数变形。

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(1) $\displaystyle{\int} \frac{\mathrm{d} x}{x^{2}}$

解：
$$
\int x^{-2} \mathrm{d}x = \frac{x^{-1}}{-1} + C = -\frac{1}{x} + C.
$$

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(2) $\displaystyle{\int} x \sqrt{x} \mathrm{~d} x$

解：
$$
x\sqrt{x} = x^{1} \cdot x^{1/2} = x^{3/2},
$$
$$
\int x^{3/2} \mathrm{d}x = \frac{x^{5/2}}{5/2} + C = \frac{2}{5} x^{5/2} + C.
$$

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(3) $\displaystyle{\int} \frac{\mathrm{d} x}{\sqrt{x}}$

解：
$$
\int x^{-1/2} \mathrm{d}x = \frac{x^{1/2}}{1/2} + C = 2\sqrt{x} + C.
$$

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(4) $\displaystyle{\int} x \sqrt[3]{x} \mathrm{~d} x$

解：
$$
x \cdot x^{1/3} = x^{4/3},
$$
$$
\int x^{4/3} \mathrm{d}x = \frac{x^{7/3}}{7/3} + C = \frac{3}{7} x^{7/3} + C.
$$

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(5) $\displaystyle{\int} \frac{\mathrm{d} x}{x^{2} \sqrt{x}}$

解：
$$
\frac{1}{x^{2} \cdot x^{1/2}} = x^{-5/2},
$$
$$
\int x^{-5/2} \mathrm{d}x = \frac{x^{-3/2}}{-3/2} + C = -\frac{2}{3} x^{-3/2} + C = -\frac{2}{3x^{3/2}} + C.
$$

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(6) $\displaystyle{\int} \sqrt[m]{x^{n}} \mathrm{~d} x$

解：
$$
\sqrt[m]{x^{n}} = x^{n/m},
$$
$$
\int x^{n/m} \mathrm{d}x = \frac{x^{n/m + 1}}{n/m + 1} + C = \frac{x^{(n+m)/m}}{(n+m)/m} + C = \frac{m}{n+m} x^{(n+m)/m} + C.
$$

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(7) $\displaystyle{\int} 5 x^{3} \mathrm{~d} x$

解：
$$
5 \int x^{3} \mathrm{d}x = 5 \cdot \frac{x^{4}}{4} + C = \frac{5}{4} x^{4} + C.
$$

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(8) $\displaystyle{\int}\left(x^{2}-3 x+2\right) \mathrm{d} x$

解：
$$
\int x^{2} \mathrm{d}x - 3\int x \mathrm{d}x + 2\int \mathrm{d}x = \frac{x^{3}}{3} - \frac{3}{2}x^{2} + 2x + C.
$$

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(9) $\displaystyle{\int} \frac{\mathrm{d} h}{\sqrt{2 g h}}$（$g$ 是常数）

解：
$$
\frac{1}{\sqrt{2g}} \int h^{-1/2} \mathrm{d}h = \frac{1}{\sqrt{2g}} \cdot 2 h^{1/2} + C = \sqrt{\frac{2h}{g}} + C.
$$

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(10) $\displaystyle{\int}\left(x^{2}+1\right)^{2} \mathrm{~d} x$

解：展开
$$
(x^{2}+1)^{2} = x^{4} + 2x^{2} + 1,
$$
$$
\int (x^{4}+2x^{2}+1) \mathrm{d}x = \frac{x^{5}}{5} + \frac{2}{3}x^{3} + x + C.
$$

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(11) $\displaystyle{\int}(\sqrt{x}+1)\left(\sqrt{x^{3}}-1\right) \mathrm{d} x$

解：
$$
\sqrt{x^{3}} = x^{3/2},
$$
展开：
$$
(\sqrt{x}+1)(x^{3/2}-1) = x^{2} - \sqrt{x} + x^{3/2} - 1,
$$
即：
$$
x^{2} + x^{3/2} - x^{1/2} - 1,
$$
积分：
$$
\frac{x^{3}}{3} + \frac{x^{5/2}}{5/2} - \frac{x^{3/2}}{3/2} - x + C = \frac{x^{3}}{3} + \frac{2}{5}x^{5/2} - \frac{2}{3}x^{3/2} - x + C.
$$

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(12) $\displaystyle{\int} \frac{(1-x)^{2}}{\sqrt{x}} \mathrm{~d} x$

解：展开
$$
(1-x)^{2} = 1 - 2x + x^{2},
$$
除以 $\sqrt{x}$：
$$
x^{-1/2} - 2x^{1/2} + x^{3/2},
$$
积分：
$$
2x^{1/2} - 2\cdot\frac{2}{3}x^{3/2} + \frac{2}{5}x^{5/2} + C = 2\sqrt{x} - \frac{4}{3}x^{3/2} + \frac{2}{5}x^{5/2} + C.
$$

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(13) $\displaystyle{\int}\left(2 \mathrm{e}^{x}+\frac{3}{x}\right) \mathrm{d} x$

解：
$$
2\int e^{x} \mathrm{d}x + 3\int \frac{1}{x} \mathrm{d}x = 2e^{x} + 3\ln|x| + C.
$$

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(14) $\displaystyle{\int}\left(\frac{3}{1+x^{2}}-\frac{2}{\sqrt{1-x^{2}}}\right) \mathrm{d} x$

解：
$$
3\int \frac{1}{1+x^{2}} \mathrm{d}x - 2\int \frac{1}{\sqrt{1-x^{2}}} \mathrm{d}x = 3\arctan x - 2\arcsin x + C.
$$

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(15) $\displaystyle{\int} \mathrm{e}^{x}\left(1-\frac{\mathrm{e}^{-x}}{\sqrt{x}}\right) \mathrm{d} x$

解：展开
$$
e^{x} - \frac{1}{\sqrt{x}},
$$
积分：
$$
e^{x} - 2\sqrt{x} + C.
$$

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(16) $\displaystyle{\int} 3^{x} \mathrm{e}^{x} \mathrm{~d} x$

解：
$$
3^{x}e^{x} = (3e)^{x},
$$
$$
\int (3e)^{x} \mathrm{d}x = \frac{(3e)^{x}}{\ln(3e)} + C = \frac{3^{x}e^{x}}{1+\ln 3} + C.
$$

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(17) $\displaystyle{\int} \frac{2 \cdot 3^{x}-5 \cdot 2^{x}}{3^{x}} \mathrm{~d} x$

解：化简
$$
2 - 5\left(\frac{2}{3}\right)^{x},
$$
积分：
$$
2x - 5 \cdot \frac{(2/3)^{x}}{\ln(2/3)} + C = 2x - \frac{5(2/3)^{x}}{\ln 2 - \ln 3} + C.
$$

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(18) $\displaystyle{\int} \sec x(\sec x-\tan x) \mathrm{d} x$

解：展开
$$
\sec^{2}x - \sec x \tan x,
$$
积分：
$$
\tan x - \sec x + C.
$$

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(19) $\displaystyle{\int} \cos ^{2} \frac{x}{2} \mathrm{~d} x$

解：用半角公式
$$
\cos^{2}\frac{x}{2} = \frac{1+\cos x}{2},
$$
积分：
$$
\frac{1}{2}\int (1+\cos x) \mathrm{d}x = \frac{x}{2} + \frac{\sin x}{2} + C.
$$

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(20) $\displaystyle{\int} \frac{\mathrm{d} x}{1+\cos 2 x}$

解：
$$
1+\cos 2x = 2\cos^{2}x,
$$
$$
\int \frac{1}{2\cos^{2}x} \mathrm{d}x = \frac{1}{2}\int \sec^{2}x \mathrm{d}x = \frac{1}{2}\tan x