integral from 0 to pi/2 of a^2cos^2(t)

解

\int_{0}^{\frac{π}{2}} a^{2} cos^{2} (t) d t

\frac{π}{4} a^{2}

解答ステップ

\int_{0}^{\frac{π}{2}} a^{2} cos^{2} (t) d t

定数を除く:

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

= a^{2} \cdot \int_{0}^{\frac{π}{2}} cos^{2} (t) d t

三角関数の公式を使用して書き換える

= a^{2} \cdot \int_{0}^{\frac{π}{2}} \frac{1 + cos ( 2 t )}{2} d t

定数を除く:

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

= a^{2} \frac{1}{2} \cdot \int_{0}^{\frac{π}{2}} 1 + cos (2 t) d t

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

= a^{2} \frac{1}{2} (\int_{0}^{\frac{π}{2}} 1 d t + \int_{0}^{\frac{π}{2}} cos (2 t) d t)

\int_{0}^{\frac{π}{2}} 1 d t = \frac{π}{2}

\int_{0}^{\frac{π}{2}} cos (2 t) d t = 0

= a^{2} \frac{1}{2} (\frac{π}{2} + 0)

簡素化

a^{2} \frac{1}{2} (\frac{π}{2} + 0) : \frac{π}{4} a^{2}

= \frac{π}{4} a^{2}