integral from x to 1 of sin^2(x)

解

\int_{x}^{1} sin^{2} (x) d x

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

解答ステップ

\int_{x}^{1} sin^{2} (x) d x

三角関数の公式を使用して書き換える

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

\int_{x}^{1} 1 d x = 1 - x

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

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