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^6(3x)cos^2(3x)

解答

\int sin^{6} (3 x) cos^{2} (3 x) d x

解答

\frac{1}{2304} (90 x - 20 sin^{3} (3 x) cos (3 x) - 15 sin (6 x) - 16 sin^{5} (3 x) cos (3 x) + 96 sin^{7} (3 x) cos (3 x)) + C

求解步骤

\int sin^{6} (3 x) cos^{2} (3 x) d x

使用换元积分法

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

提出常数:

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

= \frac{1}{3} \cdot \int sin^{6} (u) cos^{2} (u) d u

使用三角恒等式改写

= \frac{1}{3} \cdot \int sin^{6} (u) (1 - sin^{2} (u)) d u

乘开

sin^{6} (u) (1 - sin^{2} (u)) : sin^{6} (u) - sin^{8} (u)

= \frac{1}{3} \cdot \int sin^{6} (u) - sin^{8} (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}{3} (\int sin^{6} (u) d u - \int sin^{8} (u) d u)

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

\int sin^{8} (u) d u = - \frac{cos ( u ) sin ^{7} ( u )}{8} + \frac{7}{8} (- \frac{cos ( u ) sin ^{5} ( u )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u))))

= \frac{1}{3} (- \frac{cos ( u ) sin ^{5} ( u )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u))) - (- \frac{cos ( u ) sin ^{7} ( u )}{8} + \frac{7}{8} (- \frac{cos ( u ) sin ^{5} ( u )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u))))))

u = 3 x

代回

= \frac{1}{3} (- \frac{cos ( 3 x ) sin ^{5} ( 3 x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (3 x) cos (3 x) + \frac{3}{8} (3 x - \frac{1}{2} sin (2 \cdot 3 x))) - (- \frac{cos ( 3 x ) sin ^{7} ( 3 x )}{8} + \frac{7}{8} (- \frac{cos ( 3 x ) sin ^{5} ( 3 x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (3 x) cos (3 x) + \frac{3}{8} (3 x - \frac{1}{2} sin (2 \cdot 3 x))))))

化简

\frac{1}{3} (- \frac{cos ( 3 x ) sin ^{5} ( 3 x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (3 x) cos (3 x) + \frac{3}{8} (3 x - \frac{1}{2} sin (2 \cdot 3 x))) - (- \frac{cos ( 3 x ) sin ^{7} ( 3 x )}{8} + \frac{7}{8} (- \frac{cos ( 3 x ) sin ^{5} ( 3 x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (3 x) cos (3 x) + \frac{3}{8} (3 x - \frac{1}{2} sin (2 \cdot 3 x)))))) : \frac{1}{2304} (90 x - 20 sin^{3} (3 x) cos (3 x) - 15 sin (6 x) - 16 sin^{5} (3 x) cos (3 x) + 96 sin^{7} (3 x) cos (3 x))

= \frac{1}{2304} (90 x - 20 sin^{3} (3 x) cos (3 x) - 15 sin (6 x) - 16 sin^{5} (3 x) cos (3 x) + 96 sin^{7} (3 x) cos (3 x))

解答补常数

= \frac{1}{2304} (90 x - 20 sin^{3} (3 x) cos (3 x) - 15 sin (6 x) - 16 sin^{5} (3 x) cos (3 x) + 96 sin^{7} (3 x) cos (3 x)) + C

作图

流行的例子

integral of xe^4

\int x e^{4} d x

integral of e^{3x^2}*x^3

\int e^{3 x^{2}} \cdot x^{3} d x

integral of 1/(x^3*sqrt(x^2-9))

\int \frac{1}{x ^{3} \cdot x ^{2} - 9} d x

integral of (cos(y))/(1+sin(y))

\int \frac{cos ( y )}{1 + sin ( y )} d y

integral of sin^3(t)cos^6(t)

\int sin^{3} (t) cos^{6} (t) d t