integral of ((1+cos(3t))/2)

解

\int (\frac{1 + cos ( 3 t )}{2}) d t

\frac{1}{2} (t + \frac{1}{3} sin (3 t)) + C

解答ステップ

\int \frac{1 + cos ( 3 t )}{2} d t

定数を除く:

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

= \frac{1}{2} \cdot \int 1 + cos (3 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

= \frac{1}{2} (\int 1 d t + \int cos (3 t) d t)

\int 1 d t = t

\int cos (3 t) d t = \frac{1}{3} sin (3 t)

= \frac{1}{2} (t + \frac{1}{3} sin (3 t))

定数を解答に追加する

= \frac{1}{2} (t + \frac{1}{3} sin (3 t)) + C