integral of sqrt(x)(3-5x)

解答

\int x (3 - 5 x) d x

2 x^{\frac{3}{2}} - 2 x^{\frac{5}{2}} + C

求解步骤

\int x (3 - 5 x) d x

乘开

x (3 - 5 x) : 3 x - 5 x^{\frac{3}{2}}

= \int 3 x - 5 x^{\frac{3}{2}} 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 3 x d x - \int 5 x^{\frac{3}{2}} d x

\int 3 x d x = 2 x^{\frac{3}{2}}

\int 5 x^{\frac{3}{2}} d x = 2 x^{\frac{5}{2}}

= 2 x^{\frac{3}{2}} - 2 x^{\frac{5}{2}}

解答补常数

= 2 x^{\frac{3}{2}} - 2 x^{\frac{5}{2}} + C

\frac{d}{d x} (cos^{3} (e^{- x}))

\int cos (\frac{mπ x}{3}) d x

\frac{\partial}{\partial y} (2 y cos (x))

y^{^{''}} + 2 y^{^{'}} + 2 y = sin (2 x)

x \to 0 + lim (e^{- \frac{5}{x}})