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

解

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

\frac{1}{4} sin^{3} (x) cos (x) - \frac{3}{32} sin (4 x) + \frac{3}{16} sin (2 x) + cos^{3} (x) sin (x) + C

解答ステップ

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

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

\int sin^{4} (x) d x = - \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))

= cos^{3} (x) sin (x) + \frac{3}{8} (x - \frac{1}{4} sin (4 x)) - (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x)))

簡素化

cos^{3} (x) sin (x) + \frac{3}{8} (x - \frac{1}{4} sin (4 x)) - (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))) : \frac{1}{4} sin^{3} (x) cos (x) - \frac{3}{32} sin (4 x) + \frac{3}{16} sin (2 x) + cos^{3} (x) sin (x)

= \frac{1}{4} sin^{3} (x) cos (x) - \frac{3}{32} sin (4 x) + \frac{3}{16} sin (2 x) + cos^{3} (x) sin (x)

定数を解答に追加する

= \frac{1}{4} sin^{3} (x) cos (x) - \frac{3}{32} sin (4 x) + \frac{3}{16} sin (2 x) + cos^{3} (x) sin (x) + C