integral of-x^2y^2sin(x)+2xy^2cos(x)

解答

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

y^{2} x^{2} cos (x) + C

求解步骤

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

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

\int sin (x) y^{2} x^{2} d x = y^{2} (- x^{2} cos (x) + 2 (x sin (x) + cos (x)))

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

= - y^{2} (- x^{2} cos (x) + 2 (x sin (x) + cos (x))) + 2 y^{2} (x sin (x) + cos (x))

化简

- y^{2} (- x^{2} cos (x) + 2 (x sin (x) + cos (x))) + 2 y^{2} (x sin (x) + cos (x)) : y^{2} x^{2} cos (x)

= y^{2} x^{2} cos (x)

解答补常数

= y^{2} x^{2} cos (x) + C