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人気のある微分積分 >

integral of (sin^2(2x))(cos^4(2x))

解

\int (sin^{2} (2 x)) (cos^{4} (2 x)) d x

解

\frac{1}{192} (12 x + 76 cos^{3} (2 x) sin (2 x) - 9 sin (8 x) - 16 cos^{5} (2 x) sin (2 x) - 15 sin (4 x)) + C

解答ステップ

\int sin^{2} (2 x) cos^{4} (2 x) d x

u置換積分法を適用する

= \int sin^{2} (u) cos^{4} (u) \frac{1}{2} d u

定数を除く:

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

= \frac{1}{2} \cdot \int sin^{2} (u) cos^{4} (u) d u

三角関数の公式を使用して書き換える

= \frac{1}{2} \cdot \int (1 - cos^{2} (u)) cos^{4} (u) d u

拡張

(1 - cos^{2} (u)) cos^{4} (u) : cos^{4} (u) - cos^{6} (u)

= \frac{1}{2} \cdot \int cos^{4} (u) - cos^{6} (u) d u

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

= \frac{1}{2} (\int cos^{4} (u) d u - \int cos^{6} (u) d u)

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

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

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

代用を戻す

u = 2 x

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

簡素化

\frac{1}{2} (cos^{3} (2 x) sin (2 x) + \frac{3}{8} (2 x - \frac{1}{4} sin (4 \cdot 2 x)) - (\frac{sin ( 2 x ) cos ^{5} ( 2 x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (2 x) sin (2 x) + \frac{3}{8} (2 x + \frac{1}{2} sin (2 \cdot 2 x))))) : \frac{1}{192} (12 x + 76 cos^{3} (2 x) sin (2 x) - 9 sin (8 x) - 16 cos^{5} (2 x) sin (2 x) - 15 sin (4 x))

= \frac{1}{192} (12 x + 76 cos^{3} (2 x) sin (2 x) - 9 sin (8 x) - 16 cos^{5} (2 x) sin (2 x) - 15 sin (4 x))

定数を解答に追加する

= \frac{1}{192} (12 x + 76 cos^{3} (2 x) sin (2 x) - 9 sin (8 x) - 16 cos^{5} (2 x) sin (2 x) - 15 sin (4 x)) + C

グラフ

人気の例

derivative x^2-8

d er i v a t i v e x^{2} - 8

derivative h(x)=(-3x+2)^4(-2x+3)^3

d er i v a t i v e h (x) = (- 3 x + 2)^{4} (- 2 x + 3)^{3}

derivative f(x)=-8xsin(4x^2+1)

d er i v a t i v e f (x) = - 8 x sin (4 x^{2} + 1)

integral of (x+3)/(sqrt(5-4x-x^2))

\int \frac{x + 3}{5 - 4 x - x ^{2}} d x

(d^2)/(dx^2)(((x+b)(x-a))/((x-b)))

\frac{d ^{2}}{d x ^{2}} (\frac{( x + b ) ( x - a )}{( x - b )})