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人気のある微分積分 >

integral of 16sqrt(3)cos^4(3x)sin^2(3x)

解

\int 163 cos^{4} (3 x) sin^{2} (3 x) d x

解

\frac{1}{6 3} (18 x + 76 cos^{3} (3 x) sin (3 x) - 9 sin (12 x) - 16 cos^{5} (3 x) sin (3 x) - 15 sin (6 x)) + C

解答ステップ

\int 163 cos^{4} (3 x) sin^{2} (3 x) d x

定数を除く:

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

= 163 \cdot \int cos^{4} (3 x) sin^{2} (3 x) d x

u置換積分法を適用する

= 163 \cdot \int cos^{4} (u) sin^{2} (u) \frac{1}{3} d u

定数を除く:

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

= 163 \frac{1}{3} \cdot \int cos^{4} (u) sin^{2} (u) d u

三角関数の公式を使用して書き換える

= 163 \frac{1}{3} \cdot \int cos^{4} (u) (1 - cos^{2} (u)) d u

拡張

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

= 163 \frac{1}{3} \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

= 163 \frac{1}{3} (\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)))

= 163 \frac{1}{3} (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 = 3 x

= 163 \frac{1}{3} (cos^{3} (3 x) sin (3 x) + \frac{3}{8} (3 x - \frac{1}{4} sin (4 \cdot 3 x)) - (\frac{sin ( 3 x ) cos ^{5} ( 3 x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (3 x) sin (3 x) + \frac{3}{8} (3 x + \frac{1}{2} sin (2 \cdot 3 x)))))

簡素化

163 \frac{1}{3} (cos^{3} (3 x) sin (3 x) + \frac{3}{8} (3 x - \frac{1}{4} sin (4 \cdot 3 x)) - (\frac{sin ( 3 x ) cos ^{5} ( 3 x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (3 x) sin (3 x) + \frac{3}{8} (3 x + \frac{1}{2} sin (2 \cdot 3 x))))) : \frac{1}{6 3} (18 x + 76 cos^{3} (3 x) sin (3 x) - 9 sin (12 x) - 16 cos^{5} (3 x) sin (3 x) - 15 sin (6 x))

= \frac{1}{6 3} (18 x + 76 cos^{3} (3 x) sin (3 x) - 9 sin (12 x) - 16 cos^{5} (3 x) sin (3 x) - 15 sin (6 x))

定数を解答に追加する

= \frac{1}{6 3} (18 x + 76 cos^{3} (3 x) sin (3 x) - 9 sin (12 x) - 16 cos^{5} (3 x) sin (3 x) - 15 sin (6 x)) + C

グラフ

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