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受欢迎的微积分 >

integral of (sin(x))^3(cos(x))^3

解答

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

解答

- \frac{cos ^{4} ( x )}{4} + \frac{cos ^{6} ( x )}{6} + C

求解步骤

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

化简

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

使用三角恒等式改写

= \int (1 - cos^{2} (x)) sin (x) cos^{3} (x) d x

使用换元积分法

= \int - u^{3} (1 - u^{2}) d u

乘开

- u^{3} (1 - u^{2}) : - u^{3} + u^{5}

= \int - u^{3} + u^{5} 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

= - \int u^{3} d u + \int u^{5} d u

\int u^{3} d u = \frac{u ^{4}}{4}

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

= - \frac{u ^{4}}{4} + \frac{u ^{6}}{6}

u = cos (x)

代回

= - \frac{cos ^{4} ( x )}{4} + \frac{cos ^{6} ( x )}{6}

解答补常数

= - \frac{cos ^{4} ( x )}{4} + \frac{cos ^{6} ( x )}{6} + C

作图

流行的例子

derivative e^{arcsin(x)}

d er i v a t i v e e^{a r c s i n (x)}

(dy)/(dx)=-1/(2y)

\frac{d y}{d x} = - \frac{1}{2 y}

integral from-1 to 1 of-2y^2+2

\int_{- 1}^{1} - 2 y^{2} + 2 d y

y^{''}+18y^'+80y=0

y^{^{''}} + 18 y^{^{'}} + 80 y = 0

integral from 0 to 1 of (25)/(x^5)

\int_{0}^{1} \frac{25}{x ^{5}} d x