ja

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

人気のある微分積分 >

integral of sin^2(3x)*cos^4(3x)

解

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

解

\frac{1}{288} (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 sin^{2} (3 x) cos^{4} (3 x) d x

u置換積分法を適用する

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

= \frac{1}{3} \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}{3} \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}{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

= \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)))

= \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

= \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

\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}{288} (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}{288} (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}{288} (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

人気の例

integral of (2x^2-5x-18)/(x^3+x^2-6x)

\int \frac{2 x ^{2} - 5 x - 18}{x ^{3} + x ^{2} - 6 x} d x

integral of (r^2)/(r+4)

\int \frac{r ^{2}}{r + 4} d r

integral from 0 to pi/4 of (sec^2(x))

\int_{0}^{\frac{π}{4}} (sec^{2} (x)) d x

integral of x^5cos(1-x^6)

\int x^{5} cos (1 - x^{6}) d x

limit as x approaches 0 of 5x-3

x \to 0 lim (5 x - 3)