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

解答

\int π (\frac{sin ( x )}{x})^{2} d x

π (- \frac{1}{x} + \frac{cos ^{2} ( x )}{x} + Si (2 x)) + C

求解步骤

\int π (\frac{sin ( x )}{x})^{2} d x

提出常数:

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

= π \cdot \int (\frac{sin ( x )}{x})^{2} d x

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= π \cdot \int \frac{sin ^{2} ( x )}{x ^{2}} d x

使用三角恒等式改写

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

乘开

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

= π \cdot \int \frac{1}{x ^{2}} - \frac{cos ^{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

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

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

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

= π (- \frac{1}{x} - (- \frac{cos ^{2} ( x )}{x} - Si (2 x)))

化简

= π (- \frac{1}{x} + \frac{cos ^{2} ( x )}{x} + Si (2 x))

解答补常数

= π (- \frac{1}{x} + \frac{cos ^{2} ( x )}{x} + Si (2 x)) + C