integral of sin^6(x)cos^4(x)

解

\int sin^{6} (x) cos^{4} (x) d x

- \frac{1}{6} sin^{5} (x) cos (x) + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))) - \frac{11}{10} (- \frac{1}{8} sin^{7} (x) cos (x) + \frac{7}{8} (- \frac{1}{6} sin^{5} (x) cos (x) + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))))) - \frac{1}{10} sin^{9} (x) cos (x) + C

解答ステップ

\int sin^{6} (x) cos^{4} (x) d x

三角関数の公式を使用して書き換える

= \int sin^{6} (x) (1 - sin^{2} (x))^{2} d x

拡張

sin^{6} (x) (1 - sin^{2} (x))^{2} : sin^{6} (x) - 2 sin^{8} (x) + sin^{10} (x)

= \int sin^{6} (x) - 2 sin^{8} (x) + sin^{10} (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 sin^{6} (x) d x - \int 2 sin^{8} (x) d x + \int sin^{10} (x) d x

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

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

\int sin^{10} (x) d x = - \frac{cos ( x ) sin ^{9} ( x )}{10} + \frac{9}{10} (- \frac{cos ( x ) sin ^{7} ( x )}{8} + \frac{7}{8} (- \frac{cos ( x ) sin ^{5} ( x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x)))))

= - \frac{cos ( x ) sin ^{5} ( x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))) - 2 (- \frac{cos ( x ) sin ^{7} ( x )}{8} + \frac{7}{8} (- \frac{cos ( x ) sin ^{5} ( x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))))) - \frac{cos ( x ) sin ^{9} ( x )}{10} + \frac{9}{10} (- \frac{cos ( x ) sin ^{7} ( x )}{8} + \frac{7}{8} (- \frac{cos ( x ) sin ^{5} ( x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x)))))

簡素化

- \frac{cos ( x ) sin ^{5} ( x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))) - 2 (- \frac{cos ( x ) sin ^{7} ( x )}{8} + \frac{7}{8} (- \frac{cos ( x ) sin ^{5} ( x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))))) - \frac{cos ( x ) sin ^{9} ( x )}{10} + \frac{9}{10} (- \frac{cos ( x ) sin ^{7} ( x )}{8} + \frac{7}{8} (- \frac{cos ( x ) sin ^{5} ( x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))))) : - \frac{1}{6} sin^{5} (x) cos (x) + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))) - \frac{11}{10} (- \frac{1}{8} sin^{7} (x) cos (x) + \frac{7}{8} (- \frac{1}{6} sin^{5} (x) cos (x) + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))))) - \frac{1}{10} sin^{9} (x) cos (x)

= - \frac{1}{6} sin^{5} (x) cos (x) + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))) - \frac{11}{10} (- \frac{1}{8} sin^{7} (x) cos (x) + \frac{7}{8} (- \frac{1}{6} sin^{5} (x) cos (x) + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))))) - \frac{1}{10} sin^{9} (x) cos (x)

定数を解答に追加する

= - \frac{1}{6} sin^{5} (x) cos (x) + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))) - \frac{11}{10} (- \frac{1}{8} sin^{7} (x) cos (x) + \frac{7}{8} (- \frac{1}{6} sin^{5} (x) cos (x) + \frac{5}{6} (- \frac{1}{4} sin^{3} (x) cos (x) + \frac{3}{8} (x - \frac{1}{2} sin (2 x))))) - \frac{1}{10} sin^{9} (x) cos (x) + C