integral of (cos(x)+2sin(x))^2

解

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

\frac{5}{2} x - \frac{3}{4} sin (2 x) + 2 sin^{2} (x) + C

解答ステップ

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

拡張

(cos (x) + 2 sin (x))^{2} : cos^{2} (x) + 4 cos (x) sin (x) + 4 sin^{2} (x)

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

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

\int 4 cos (x) sin (x) d x = 2 sin^{2} (x)

\int 4 sin^{2} (x) d x = 2 x - sin (2 x)

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

簡素化

\frac{1}{2} (x + \frac{1}{2} sin (2 x)) + 2 sin^{2} (x) + 2 x - sin (2 x) : \frac{5}{2} x - \frac{3}{4} sin (2 x) + 2 sin^{2} (x)

= \frac{5}{2} x - \frac{3}{4} sin (2 x) + 2 sin^{2} (x)

定数を解答に追加する

= \frac{5}{2} x - \frac{3}{4} sin (2 x) + 2 sin^{2} (x) + C