积分T
先记录一下要写的积分题，等考完试一起写解答。
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例1：求：$$\int \frac{1}{1 + e ^ {x}} \mathrm{d}x$$
\[\int \frac{1}{1 + e ^ {x}} \mathrm{d}x = \int \frac{1 + e ^ {x} - e ^ {x}}{1 + e ^ {x}} \mathrm{d}x = \int 1 - \frac{e ^ {x}}{1 + e ^ {x}} \mathrm{d}x
\]
\[= \int \mathrm{d}x - \int \frac{1}{1 + e ^ {x}} \mathrm{d}(e ^ {x} + 1) = x - \ln(e ^ {x} + 1) + C
\]
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例2：求：$$\int \frac{1}{x(1 + x ^ {6})}\mathrm{d}x$$
\[\int \frac{1}{x(1 + x ^ {6})} \mathrm{d}x = \int \frac{x ^ {5}}{x ^ {6}(1 + x ^ {6})} \mathrm{d}x = \frac{1}{6} \int \frac{1}{x ^ {6}(1 + x ^ {6})} \mathrm{d}x ^ {6}
\]
\[= \frac{1}{6} \int \frac{1}{x ^ {6}} - \frac{1}{1 + x ^ {6}} \mathrm{d}x ^ {6} = \frac{1}{6} \int \frac{1}{x ^ {6}}\mathrm{d}x ^ {6} - \int \frac{1}{1 + x ^ {6}} \mathrm{d}(x ^ {6} + 1)
\]
\[= \frac{1}{6}(\ln x ^ {6} - \ln x ^ {6} + 1) + C= \frac{1}{6} \ln \frac{x ^ {6}}{1 + x ^ {6}} + C
\]
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例3：求：$$\int \frac{e ^ {2x}}{1 + e ^ {x}}\mathrm{d}x$$
\[= \int \frac{e ^ {x}}{1 + e ^ {x}} \mathrm{d}e ^ {x} = \int \frac{1 + e ^ {x} - 1}{1 + e ^ {x}} \mathrm{d}e ^ {x} = \int 1 - \frac{1}{1 + e ^ {x}} \mathrm{d}e ^ {x}
\]
\[= e ^ {x} - \ln(e ^ {x} + 1) + C
\]
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例4：求$$\int \tan ^ {4} x\mathrm{d}x$$
\[= \int \tan ^ {2} x \left( \frac{1}{\cos ^ {2} x} - 1\right) \mathrm{d}x = \int \tan ^ {2} x \frac{\mathrm{d}x}{\cos ^ {2} x} - \int \tan ^ {2} x\mathrm{d}x
\]
\[= \int \tan ^ {2}x \mathrm{d}\tan x - \int \frac{1 -\cos ^ {2}} {\cos ^ {2} x}\mathrm{d}x = \int \tan ^ {2} x \mathrm{d} \tan x - \int \frac{1}{\cos ^ {2} x} \mathrm{d}x + \int \mathrm{d}x
\]
\[= \frac{1}{3} \tan ^ {3} x - \tan x + x + C
\]
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例5：$$\int \frac{1}{a ^ {2} \cos ^ {2} x + b ^ {2} \sin ^ {2} x}\mathrm{d}x$$
\[\int \frac{\frac{\cos ^ {2} x}{\cos ^ {2} x}}{a ^ {2} \cos ^ {2}x + b ^ {2} \sin ^ {2}x} \mathrm{d}x = \int \frac{\frac{1}{\cos ^ {2} x}}{a ^ {2} + b ^ {2} \tan ^ {2} x} \mathrm{d}x = \int \frac{1}{a ^ {2} + b ^ {2} \tan ^ {2} x} \mathrm{d} \tan x
\]
\[= \frac{1}{a ^ {2}} \int \frac{1}{1 + (\frac{b}{a}\tan x ) ^ {2}} \mathrm{d} \tan x = = \frac{b}{a} \int \frac{1}{1 + (\frac{b}{a} \tan x) ^ {2}} \mathrm{d} \frac{b}{a} \tan x
\]
\[= \frac{b}{a} \arctan \frac{b}{a} \tan x + C
\]
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例6：求：$$\int \frac{x + 1}{x ^ {2} - 3x + 1}\mathrm{d}x$$
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例7：求：$$\int \frac{\ln \tan x}{\sin x \cos x}\mathrm{d}x$$
\[= \int \ln \tan x \ \mathrm{d} (\ln \tan x) = \frac{1}{2} \left( \ln \tan x\right) ^ {2} + C
\]
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例8：求：$$\int \frac{1}{\sin 2x \cos x} \mathrm{d}x$$
\[\int \frac{1}{2 \sin x \cos ^ {2} x} \mathrm{d}x = \int \frac{\sin ^ {2} x + \cos ^ {2} x}{2 \sin x \cos ^ {2} x} \mathrm{d}x = \int \frac{1 + \tan ^ {2} x}{ 2 \sin x} \mathrm{d}x
\]
\[= \int \frac{1}{2 \sin x} \mathrm{d}x + \int \frac{\sin x}{2\cos ^ {2} x} \mathrm{d} x = \frac{1}{2} \int \frac{\sin x}{1 - \cos ^ {2} x} \mathrm{d}x - \frac{1}{2} \int \frac{1}{\cos ^ {2} x} \mathrm{d} \cos x
\]
\[= -\frac{1}{2} \int \frac{1}{(1 + \cos x)(1 - \cos x)} \mathrm{d} \cos x - \frac{1}{2}(- \frac{1}{ \cos x})
\]
\[= \frac{1}{4}\int \frac{1}{\cos x - 1} - \frac{1}{ \cos x + 1} \mathrm{d} \cos x + \frac{1}{2\cos x}
\]
\[= \frac{1}{4} \ln | \frac{\cos x - 1}{\cos x + 1}| + \frac{1}{2 \cos x} + C
\]
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例9：求：$$\int \frac{\sin ^ {5} x}{\cos ^ {8} x} \mathrm{d}x$$
\[= \int \frac{(1 - \cos ^ {2} x) ^ {2} \sin x \mathrm{d}x}{\cos ^ {8} x} = -\int \frac{1 - 2 \cos ^ {2} x + \cos ^ {4} x}{\cos ^ {8} x} \mathrm{d} \cos x
\]
\[= \int - \frac{1}{\cos ^ {8} x} \mathrm{d} \cos x + \int \frac{2}{\cos ^ {6} x }\mathrm{d} \cos x - \int \frac{1}{\cos ^ {4} x} \mathrm{d} \cos x
\]
\[= \frac{1}{7\cos ^ {7} x} - \frac{2}{5 \cos ^ {5} x} + \frac{1}{3 \cos ^ {3} x} + C
\]
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例10：求：$$\int \frac{\sin x}{1 + \sin x} \mathrm{d}x$$
\[= \int \frac{1 + \sin x - 1}{1 + \sin x} \mathrm{d}x = \int 1 - \frac{1}{1 + \sin x} \mathrm{d}x = \int \mathrm{d}x - \int \frac{\mathrm{d}x}{1 + \sin x}
\]
\[=\int \mathrm{d}x - \int \frac{1 - \sin x}{\cos ^ {2} x} \mathrm{d} x = \int \mathrm{d}x - \int \frac{1}{\cos ^ {2} x} \mathrm{d}x + \int \frac{\sin x}{\cos ^ {2} x} \mathrm{d}x
\]
\[ = x -\tan x - \int \frac{\mathrm{d} \cos x}{\cos ^ {2} x} + C = x - \tan x
+ \frac{1}{\cos x} + C_{1}\]
