integral of cos^8(x)

解

\int cos^{8} (x) d x

\frac{sin ( x ) cos ^{7} ( x )}{8} + \frac{7}{8} (\frac{sin ( x ) cos ^{5} ( x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (x) sin (x) + \frac{3}{8} (x + \frac{1}{2} sin (2 x)))) + C

解答ステップ

\int cos^{8} (x) d x

積分換算を適用する

= \frac{sin ( x ) cos ^{7} ( x )}{8} + \frac{7}{8} \cdot \int cos^{6} (x) d x

\int cos^{6} (x) d x = \frac{sin ( x ) cos ^{5} ( x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (x) sin (x) + \frac{3}{8} (x + \frac{1}{2} sin (2 x)))

= \frac{sin ( x ) cos ^{7} ( x )}{8} + \frac{7}{8} (\frac{sin ( x ) cos ^{5} ( x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (x) sin (x) + \frac{3}{8} (x + \frac{1}{2} sin (2 x))))

定数を解答に追加する

= \frac{sin ( x ) cos ^{7} ( x )}{8} + \frac{7}{8} (\frac{sin ( x ) cos ^{5} ( x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (x) sin (x) + \frac{3}{8} (x + \frac{1}{2} sin (2 x)))) + C