zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的微积分 >

integral of sin^2(5x)cos^4(5x)

解答

\int sin^{2} (5 x) cos^{4} (5 x) d x

解答

\frac{1}{480} (30 x + 76 cos^{3} (5 x) sin (5 x) - 9 sin (20 x) - 16 cos^{5} (5 x) sin (5 x) - 15 sin (10 x)) + C

求解步骤

\int sin^{2} (5 x) cos^{4} (5 x) d x

使用换元积分法

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

提出常数:

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

= \frac{1}{5} \cdot \int sin^{2} (u) cos^{4} (u) d u

使用三角恒等式改写

= \frac{1}{5} \cdot \int (1 - cos^{2} (u)) cos^{4} (u) d u

乘开

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

= \frac{1}{5} \cdot \int cos^{4} (u) - cos^{6} (u) d u

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{5} (\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}{5} (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)))))

u = 5 x

代回

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

化简

\frac{1}{5} (cos^{3} (5 x) sin (5 x) + \frac{3}{8} (5 x - \frac{1}{4} sin (4 \cdot 5 x)) - (\frac{sin ( 5 x ) cos ^{5} ( 5 x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (5 x) sin (5 x) + \frac{3}{8} (5 x + \frac{1}{2} sin (2 \cdot 5 x))))) : \frac{1}{480} (30 x + 76 cos^{3} (5 x) sin (5 x) - 9 sin (20 x) - 16 cos^{5} (5 x) sin (5 x) - 15 sin (10 x))

= \frac{1}{480} (30 x + 76 cos^{3} (5 x) sin (5 x) - 9 sin (20 x) - 16 cos^{5} (5 x) sin (5 x) - 15 sin (10 x))

解答补常数

= \frac{1}{480} (30 x + 76 cos^{3} (5 x) sin (5 x) - 9 sin (20 x) - 16 cos^{5} (5 x) sin (5 x) - 15 sin (10 x)) + C

作图

流行的例子

integral from-3 to 3 of 9x^2

\int_{- 3}^{3} 9 x^{2} d x

y^{''}-6y^'+9y=6x^2e^{3x}

y^{^{''}} - 6 y^{^{'}} + 9 y = 6 x^{2} e^{3 x}

integral of 1/(10x^4)

\int \frac{1}{10 x ^{4}} d x

derivative of (10/((x+5)^2))

\frac{d}{d x} (\frac{10}{( x + 5 ) ^{2}})

泰勒 x*sin(x)

t a y l or x \cdot sin (x)