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人気のある微分積分 >

integral of 1024sin^4(x)cos^4(x)

解

\int 1024 sin^{4} (x) cos^{4} (x) d x

解

1024 (- \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 1024 sin^{4} (x) cos^{4} (x) d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 1024 \cdot \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

= 1024 \cdot \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)

= 1024 \cdot \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

= 1024 (\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))))

= 1024 (- \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

1024 (- \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))))) : 1024 (- \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))

= 1024 (- \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

= 1024 (- \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

(\partial)/(\partial z)(e^{xyz^2+z})

\frac{\partial}{\partial z} (e^{x y z^{2} + z})

(\partial)/(\partial x)(4x^2+4y^3)

\frac{\partial}{\partial x} (4 x^{2} + 4 y^{3})

integral of (3x^2+4)sqrt(x-1)

\int (3 x^{2} + 4) x - 1 d x

integral from 4 to 7 of (x-5)^2

\int_{4}^{7} (x - 5)^{2} d x

y^'+y=e^x,y(0)=6

y^{^{'}} + y = e^{x}, y (0) = 6