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人気のある微分積分 >

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

解

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

解

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

解答ステップ

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

簡素化

= \int sin^{4} (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^{4} (x) (1 - sin^{2} (x))^{2} d x

拡張

sin^{4} (x) (1 - sin^{2} (x))^{2} : sin^{4} (x) - 2 sin^{6} (x) + sin^{8} (x)

= \int sin^{4} (x) - 2 sin^{6} (x) + sin^{8} (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^{4} (x) d x - \int 2 sin^{6} (x) d x + \int sin^{8} (x) d 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))

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

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

定数を解答に追加する

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

グラフ

人気の例

integral of 1/(xsqrt(49x^2-36))

\int \frac{1}{x 49 x ^{2} - 36} d x

integral of |t^2-1|

\int t^{2} - 1 d x

integral of 2/(xsqrt(4-x^2))

\int \frac{2}{x 4 - x ^{2}} d x

integral of 1/(2-3sin(x))

\int \frac{1}{2 - 3 sin ( x )} d x

integral of ln(2w)

\int ln (2 w) d w