integral of xsin(3+3x)

解答

\int x sin (3 + 3 x) d x

\frac{1}{9} (- 3 x cos (3 + 3 x) + sin (3 + 3 x)) + C

求解步骤

\int x sin (3 + 3 x) d x

使用换元积分法

= \int \frac{sin ( u ) ( u - 3 )}{9} d u

提出常数:

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

= \frac{1}{9} \cdot \int sin (u) (u - 3) d u

使用分布积分法

= \frac{1}{9} (- cos (u) (u - 3) - \int - cos (u) d u)

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

= \frac{1}{9} (- cos (u) (u - 3) - (- sin (u)))

u = 3 + 3 x

代回

= \frac{1}{9} (- cos (3 + 3 x) (3 + 3 x - 3) - (- sin (3 + 3 x)))

化简

\frac{1}{9} (- cos (3 + 3 x) (3 + 3 x - 3) - (- sin (3 + 3 x))) : \frac{1}{9} (- 3 x cos (3 + 3 x) + sin (3 + 3 x))

= \frac{1}{9} (- 3 x cos (3 + 3 x) + sin (3 + 3 x))

解答补常数

= \frac{1}{9} (- 3 x cos (3 + 3 x) + sin (3 + 3 x)) + C

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

x \to 0 lim ((1 - 2 x)^{(\frac{1}{x})})

\frac{d}{d x} (ln (10 x^{2} - 8))

y^{^{'}} = e^{2 t + y - 1} - 2

\frac{\partial}{\partial x} (e^{x y} + ln (\frac{x}{y}))