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^7(2x)cos^5(2x)

解答

\int sin^{7} (2 x) cos^{5} (2 x) d x

解答

\frac{1}{2} (- \frac{cos ^{6} ( 2 x )}{6} + \frac{3 cos ^{8} ( 2 x )}{8} - \frac{3 cos ^{10} ( 2 x )}{10} + \frac{cos ^{12} ( 2 x )}{12}) + C

求解步骤

\int sin^{7} (2 x) cos^{5} (2 x) d x

使用换元积分法

= \int sin^{7} (u) cos^{5} (u) \frac{1}{2} d u

提出常数:

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

= \frac{1}{2} \cdot \int sin^{7} (u) cos^{5} (u) d u

使用三角恒等式改写

= \frac{1}{2} \cdot \int (1 - cos^{2} (u))^{3} sin (u) cos^{5} (u) d u

使用换元积分法

= \frac{1}{2} \cdot \int - v^{5} (1 - v^{2})^{3} d v

乘开

- v^{5} (1 - v^{2})^{3} : - v^{5} + 3 v^{7} - 3 v^{9} + v^{11}

= \frac{1}{2} \cdot \int - v^{5} + 3 v^{7} - 3 v^{9} + v^{11} d v

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}{2} (- \int v^{5} d v + \int 3 v^{7} d v - \int 3 v^{9} d v + \int v^{11} d v)

\int v^{5} d v = \frac{v ^{6}}{6}

\int 3 v^{7} d v = \frac{3 v ^{8}}{8}

\int 3 v^{9} d v = \frac{3 v ^{10}}{10}

\int v^{11} d v = \frac{v ^{12}}{12}

= \frac{1}{2} (- \frac{v ^{6}}{6} + \frac{3 v ^{8}}{8} - \frac{3 v ^{10}}{10} + \frac{v ^{12}}{12})

代回

= \frac{1}{2} (- \frac{cos ^{6} ( 2 x )}{6} + \frac{3 cos ^{8} ( 2 x )}{8} - \frac{3 cos ^{10} ( 2 x )}{10} + \frac{cos ^{12} ( 2 x )}{12})

解答补常数

= \frac{1}{2} (- \frac{cos ^{6} ( 2 x )}{6} + \frac{3 cos ^{8} ( 2 x )}{8} - \frac{3 cos ^{10} ( 2 x )}{10} + \frac{cos ^{12} ( 2 x )}{12}) + C

作图

流行的例子

integral of 1/(1+sec(3x))

\int \frac{1}{1 + sec ( 3 x )} d x

integral of 1/(cos(z))

\int \frac{1}{cos ( z )} d z

integral of x+sin(y)

\int x + sin (y) d x

integral of ((3-x))/(16+6x-x^2)

\int \frac{( 3 - x )}{16 + 6 x - x ^{2}} d x

integral of (9x+15)/(x^2+2x+1)

\int \frac{9 x + 15}{x ^{2} + 2 x + 1} d x