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受欢迎的微积分 >

integral of (x(sin(x^2)))(cos^8(x^2))

解答

\int (x (sin (x^{2}))) (cos^{8} (x^{2})) d x

解答

- \frac{1}{18} cos^{9} (x^{2}) + C

求解步骤

\int x sin (x^{2}) cos^{8} (x^{2}) d x

使用换元积分法

= \int \frac{cos ^{8} ( u ) sin ( u )}{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 cos^{8} (u) sin (u) d u

使用换元积分法

= \frac{1}{2} \cdot \int - v^{8} d v

提出常数:

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

= \frac{1}{2} (- \int v^{8} d v)

使用幂法则

= \frac{1}{2} (- \frac{v ^{9}}{9})

代回

= \frac{1}{2} (- \frac{cos ^{9} ( x ^{2} )}{9})

化简

\frac{1}{2} (- \frac{cos ^{9} ( x ^{2} )}{9}) : - \frac{1}{18} cos^{9} (x^{2})

= - \frac{1}{18} cos^{9} (x^{2})

解答补常数

= - \frac{1}{18} cos^{9} (x^{2}) + C

作图

流行的例子

derivative of (3x-1^4(-2x+9)^5)

\frac{d}{d x} ((3 x - 1)^{4} (- 2 x + 9)^{5})

y^{''}-7y+10y=0

y^{^{''}} - 7 y + 10 y = 0

derivative of 8/(x^9)

\frac{d}{d x} (\frac{8}{x ^{9}})

derivative of sqrt(7x+9)

\frac{d}{d x} (7 x + 9)

derivative f(t)=(1+sqrt(t))(2t^2-7)

d er i v a t i v e f (t) = (1 + t) (2 t^{2} - 7)