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

解

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

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

解答ステップ

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

指数の規則を適用する:

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

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

三角関数の公式を使用して書き換える

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

拡張

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

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

総和規則を適用する:

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

= \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)

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

簡素化

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

定数を解答に追加する

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