人邮高数 第3章 第3-1-2题

[AI解答]

[AI解答]

以下为各题的不定积分求解过程，均使用基本积分公式与代数变形。

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(1)
$$
\int (x^2 + 5\sqrt{x} + \ln 3) \, dx
= \frac{x^3}{3} + 5 \cdot \frac{2}{3} x^{3/2} + x\ln 3 + C
= \frac{x^3}{3} + \frac{10}{3} x^{3/2} + x\ln 3 + C
$$

(2)
$$
\int x\sqrt{x} \, dx = \int x^{3/2} dx = \frac{2}{5} x^{5/2} + C
$$

(3)
$$
\int \left(1 + \frac{1}{x}\right) dx = x + \ln|x| + C
$$

(4)
$$
\int \left(\frac{2}{x^2} + \frac{3}{1+x^2}\right) dx
= \int 2x^{-2} dx + 3\int \frac{dx}{1+x^2}
= -\frac{2}{x} + 3\arctan x + C
$$

(5)
$$
\int \left(x^2 + 2^x + \frac{2}{x}\right) dx
= \frac{x^3}{3} + \frac{2^x}{\ln 2} + 2\ln|x| + C
$$

(6)
$$
\int \left(a^{2/3} + x^{2/3}\right) dx
= a^{2/3}x + \frac{3}{5} x^{5/3} + C
$$

(7)
$$
\int \left(2x^3 - \sin x + 5\sqrt{x}\right) dx
= \frac{1}{2}x^4 + \cos x + \frac{10}{3}x^{3/2} + C
$$

(8)
$$
\int \frac{\sqrt{x} - x + x^2 e^x}{x^2} dx
= \int \left(x^{-3/2} - \frac{1}{x} + e^x\right) dx
= -2x^{-1/2} - \ln|x| + e^x + C
$$

(9)
注意到 $(\sqrt{x}+1)(x - \sqrt{x} + 1) = x^{3/2} + 1$
$$
\int (x^{3/2} + 1) dx = \frac{2}{5} x^{5/2} + x + C
$$

(10)
$$
\left(1 - x^{2/3}\right)^2 = 1 - 2x^{2/3} + x^{4/3}
$$
$$
\int \left(1 - 2x^{2/3} + x^{4/3}\right) dx
= x - 2\cdot\frac{3}{5}x^{5/3} + \frac{3}{7}x^{7/3} + C
= x - \frac{6}{5}x^{5/3} + \frac{3}{7}x^{7/3} + C
$$

(11)
$$
\frac{x^2}{1+x^2} = 1 - \frac{1}{1+x^2}
$$
$$
\int \left(1 - \frac{1}{1+x^2}\right) dx = x - \arctan x + C
$$

(12)
$$
\frac{x^2}{1-x^2} = -1 + \frac{1}{1-x^2}
$$
$$
\int \left(-1 + \frac{1}{1-x^2}\right) dx = -x + \frac{1}{2}\ln\left|\frac{1+x}{1-x}\right| + C
$$

(13)
$$
\frac{1+2x^2}{x^2(1+x^2)} = \frac{1}{x^2} + \frac{1}{1+x^2}
$$
$$
\int \left(\frac{1}{x^2} + \frac{1}{1+x^2}\right) dx = -\frac{1}{x} + \arctan x + C
$$

(14)
$$
\frac{3x^4+3x^2+1}{x^2+1} = 3x^2 + \frac{1}{x^2+1}
$$
$$
\int \left(3x^2 + \frac{1}{1+x^2}\right) dx = x^3 + \arctan x + C
$$

(15)
$$
\frac{1+x}{\sqrt[3]{x}} = x^{-1/3} + x^{2/3}
$$
$$
\int \left(x^{-1/3} + x^{2/3}\right) dx = \frac{3}{2}x^{2/3} + \frac{3}{5}x^{5/3} + C
$$

(16)
化简被积函数：
$$
e^{-x}\left[e^{2x} - \frac{e^x(x+1)}{x^2}\right] = e^x - \frac{x+1}{x^2}
$$
$$
\int \left(e^x - \frac{1}{x} - \frac{1}{x^2}\right) dx = e^x - \ln|x| + \frac{1}{x} + C
$$

(17)
$$
(2^x + 3^x)^2 = 4^x + 9^x + 2\cdot 6^x
$$
$$
\int (4^x + 9^x + 2\cdot 6^x) dx = \frac{4^x}{\ln 4} + \frac{9^x}{\ln 9} + \frac{2\cdot 6^x}{\ln 6} + C
$$

(18)
$$
\frac{2\cdot 3^x - 5\cdot 2^x}{3^x} = 2 - 5\left(\frac{2}{3}\right)^x
$$
$$
\int \left[2 - 5\left(\frac{2}{3}\right)^x\right] dx = 2x - \frac{5\left(\frac{2}{3}\right)^x}{\ln(2/3)} + C
$$

(19)
$$
\sqrt{x\sqrt{x}} = x^{3/4}
$$
$$
\left(1 - \frac{1}{x^2}\right)x^{3/4} = x^{3/4} - x^{-5/4}
$$
$$
\int \left(x^{3/4} - x^{-5/4}\right) dx = \frac{4}{7}x^{7/4} + 4x^{-1/4} + C
$$

(20)
$$
\int \left(\frac{3}{1+x^2} - \frac{2}{\sqrt{1-x^2}}\right) dx = 3\arctan x - 2\arcsin x + C
$$

(21)
$$
\frac{\cos 2x}{\cos x - \sin x} = \frac{\cos^2 x - \sin^2 x}{\cos x - \sin x} = \cos x + \sin x
$$
$$
\int (\cos x + \sin x) dx = \sin x - \cos x + C
$$

(22)
$$
\frac{1+\cos^2 x}{1+\cos 2x} = \frac{1+\cos^2 x}{2\cos^2 x} = \frac{1}{2\cos^2 x} + \frac{1}{2}
$$
$$
\int \left(\frac{1}{2}\sec^2 x + \frac12\right) dx = \frac12 \tan x + \frac{x}{2} + C
$$

(23)
$$
\left(\sin\frac{x}{2} + \cos\frac{x}{2}\right)^2 = 1 + \sin x
$$
$$
\int (1 + \sin x) dx = x - \cos x + C
$$

(24)
$$
\frac{1}{\cos^2 x \sin^2 x} = \frac{1}{\sin^2 x} \cdot \frac{1}{\cos^2 x} = \csc^2 x \sec^2 x
$$
或利用 $\sin 2x = 2\sin x\cos x$：
$$
\frac{1}{\cos^2 x \sin^2 x} = \frac{4}{\sin^2 2x} = 4\csc^2 2x
$$
$$
\int 4\csc^2 2x \, dx = -2\cot 2x + C
$$

(25)
$$
\int e^{x+2} dx = e^2 \int e^x dx = e^{x+2} + C
$$

(26)
$$
3^{2x} e^x = (9e)^x