integral of ((sin^2(1/x)))/(x^2)

解答

\int \frac{( sin ^{2} ( \frac{1}{x} ) )}{x ^{2}} d x

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

求解步骤

\int \frac{sin ^{2} ( \frac{1}{x} )}{x ^{2}} d x

使用三角恒等式改写

= \int \frac{1 - cos ( \frac{2}{x} )}{2 x ^{2}} d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{1 - cos ( \frac{2}{x} )}{x ^{2}} d x

乘开

\frac{1 - cos ( \frac{2}{x} )}{x ^{2}} : \frac{1}{x ^{2}} - \frac{cos ( \frac{2}{x} )}{x ^{2}}

= \frac{1}{2} \cdot \int \frac{1}{x ^{2}} - \frac{cos ( \frac{2}{x} )}{x ^{2}} d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int \frac{1}{x ^{2}} d x - \int \frac{cos ( \frac{2}{x} )}{x ^{2}} d x)

\int \frac{1}{x ^{2}} d x = - \frac{1}{x}

\int \frac{cos ( \frac{2}{x} )}{x ^{2}} d x = - \frac{1}{2} sin (\frac{2}{x})

= \frac{1}{2} (- \frac{1}{x} - (- \frac{1}{2} sin (\frac{2}{x})))

化简

= \frac{1}{2} (- \frac{1}{x} + \frac{1}{2} sin (\frac{2}{x}))

解答补常数

= \frac{1}{2} (- \frac{1}{x} + \frac{1}{2} sin (\frac{2}{x})) + C

\int (\frac{1 - 4 x ^{4} + x ^{3}}{x ^{3}}) d x

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