integral of cos(x)-cos^4(x)

解

\int cos (x) - cos^{4} (x) d x

sin (x) - cos^{3} (x) sin (x) - \frac{3}{8} x + \frac{3}{32} sin (4 x) + C

解答ステップ

\int cos (x) - cos^{4} (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 (x) d x - \int cos^{4} (x) d x

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

\int cos^{4} (x) d x = cos^{3} (x) sin (x) + \frac{3}{8} (x - \frac{1}{4} sin (4 x))

= sin (x) - (cos^{3} (x) sin (x) + \frac{3}{8} (x - \frac{1}{4} sin (4 x)))

簡素化

sin (x) - (cos^{3} (x) sin (x) + \frac{3}{8} (x - \frac{1}{4} sin (4 x))) : sin (x) - cos^{3} (x) sin (x) - \frac{3}{8} x + \frac{3}{32} sin (4 x)

= sin (x) - cos^{3} (x) sin (x) - \frac{3}{8} x + \frac{3}{32} sin (4 x)

定数を解答に追加する

= sin (x) - cos^{3} (x) sin (x) - \frac{3}{8} x + \frac{3}{32} sin (4 x) + C