integral from 0 to pi/4 of cos^3(x)

解

\int_{0}^{\frac{π}{4}} cos^{3} (x) d x

\frac{5}{6 2}

十進法表記

0.58925 \dots

解答ステップ

\int_{0}^{\frac{π}{4}} cos^{3} (x) d x

三角関数の公式を使用して書き換える

= \int_{0}^{\frac{π}{4}} (1 - sin^{2} (x)) cos (x) d x

u置換積分法を適用する

= \int_{0}^{\frac{2}{2}} 1 - u^{2} d u

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_{0}^{\frac{2}{2}} 1 d u - \int_{0}^{\frac{2}{2}} u^{2} d u

\int_{0}^{\frac{2}{2}} 1 d u = \frac{1}{2}

\int_{0}^{\frac{2}{2}} u^{2} d u = \frac{1}{6 2}

= \frac{1}{2} - \frac{1}{6 2}

簡素化

\frac{1}{2} - \frac{1}{6 2} : \frac{5}{6 2}

= \frac{5}{6 2}