integral of (x-1)^2sin^2(x)

解答

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

\frac{1}{3} x^{3} sin^{2} (x) + \frac{1}{6} x^{3} cos (2 x) - \frac{1}{4} x^{2} sin (2 x) - x^{2} + \frac{x ^{2}}{2} - \frac{1}{4} x cos (2 x) + \frac{1}{2} x sin (2 x) + \frac{1}{2} x - \frac{1}{8} sin (2 x) + \frac{1}{4} cos (2 x) + C

求解步骤

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

乘开

(x - 1)^{2} sin^{2} (x) : x^{2} sin^{2} (x) - 2 x sin^{2} (x) + sin^{2} (x)

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

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

\int 2 x sin^{2} (x) d x = x^{2} - \frac{1}{2} x sin (2 x) - \frac{x ^{2}}{2} - \frac{1}{4} cos (2 x)

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

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

化简

\frac{1}{3} x^{3} sin^{2} (x) + \frac{1}{6} x^{3} cos (2 x) - \frac{1}{4} x^{2} sin (2 x) - \frac{1}{4} x cos (2 x) + \frac{1}{8} sin (2 x) - (x^{2} - \frac{1}{2} x sin (2 x) - \frac{x ^{2}}{2} - \frac{1}{4} cos (2 x)) + \frac{1}{2} (x - \frac{1}{2} sin (2 x)) : \frac{1}{3} x^{3} sin^{2} (x) + \frac{1}{6} x^{3} cos (2 x) - \frac{1}{4} x^{2} sin (2 x) - x^{2} + \frac{x ^{2}}{2} - \frac{1}{4} x cos (2 x) + \frac{1}{2} x sin (2 x) + \frac{1}{2} x - \frac{1}{8} sin (2 x) + \frac{1}{4} cos (2 x)

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

解答补常数

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

t an g e n t f (x) = \frac{9}{x ^{2}}, at x = 3

\frac{\partial}{\partial x} (e^{- 7 y} sin (7 x))

\frac{d}{d x} (\frac{4 x}{5 2} + 3)

l a pl a ce t r an s f or m 25 t^{2}

d er i v a t i v e f (t) = 7 - \frac{1}{3} t