integral of (1/2+cos(x))^2

解

\int (\frac{1}{2} + cos (x))^{2} d x

\frac{3}{4} x + \frac{1}{4} sin (2 x) + sin (x) + C

解答ステップ

\int (\frac{1}{2} + cos (x))^{2} d x

簡素化

(\frac{1}{2} + cos (x))^{2} : \frac{1}{4} + cos (x) + cos^{2} (x)

= \int \frac{1}{4} + cos (x) + 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

= \int \frac{1}{4} d x + \int cos (x) d x + \int cos^{2} (x) d x

\int \frac{1}{4} d x = \frac{1}{4} x

\int cos (x) d x = sin (x)

\int cos^{2} (x) d x = \frac{1}{2} (x + \frac{1}{2} sin (2 x))

= \frac{1}{4} x + sin (x) + \frac{1}{2} (x + \frac{1}{2} sin (2 x))

簡素化

\frac{1}{4} x + sin (x) + \frac{1}{2} (x + \frac{1}{2} sin (2 x)) : \frac{3}{4} x + \frac{1}{4} sin (2 x) + sin (x)

= \frac{3}{4} x + \frac{1}{4} sin (2 x) + sin (x)

定数を解答に追加する

= \frac{3}{4} x + \frac{1}{4} sin (2 x) + sin (x) + C