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

解

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

\frac{1}{2} (ln ∣ x ∣ - Ci (2 x)) + C

解答ステップ

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

定数を除く:

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

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

拡張

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

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{cos ( 2 x )}{x} d x = Ci (2 x)

= \frac{1}{2} (ln ∣ x ∣ - Ci (2 x))

定数を解答に追加する

= \frac{1}{2} (ln ∣ x ∣ - Ci (2 x)) + C