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受欢迎的微积分 >

integral of xysin(x^2y)

解答

\int x y sin (x^{2} y) d x

解答

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

求解步骤

\int x y sin (x^{2} y) d x

提出常数:

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

= y \cdot \int sin (y x^{2}) x d x

使用换元积分法

= y \cdot \int \frac{sin ( u )}{2 y} d u

提出常数:

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

= y \frac{1}{2 y} \cdot \int sin (u) d u

使用积分常见定则:

\int sin (u) d u = - cos (u)

= y \frac{1}{2 y} (- cos (u))

u = y x^{2}

代回

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

化简

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

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

解答补常数

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

作图

流行的例子

(\partial)/(\partial y)(2y^3)

\frac{\partial}{\partial y} (2 y^{3})

integral of (16x^2)/((x-24)(x+8)^2)

\int \frac{16 x ^{2}}{( x - 24 ) ( x + 8 ) ^{2}} d x

(\partial)/(\partial y)((x+1)/(y-1))

\frac{\partial}{\partial y} (\frac{x + 1}{y - 1})

integral from 0 to 1 of 1/(sqrt(x)(1+x))

\int_{0}^{1} \frac{1}{x ( 1 + x )} d x

integral of ysqrt(y+1)

\int y y + 1 d y