integral of (sqrt(x)+3)^3

解答

\int (x + 3)^{3} d x

\frac{2}{5} x^{\frac{5}{2}} + \frac{9 x ^{2}}{2} + 18 x^{\frac{3}{2}} + 27 x + C

求解步骤

\int (x + 3)^{3} d x

乘开

(x + 3)^{3} : x^{\frac{3}{2}} + 9 x + 27 x + 27

= \int x^{\frac{3}{2}} + 9 x + 27 x + 27 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^{\frac{3}{2}} d x + \int 9 x d x + \int 27 x d x + \int 27 d x

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

\int 9 x d x = \frac{9 x ^{2}}{2}

\int 27 x d x = 18 x^{\frac{3}{2}}

\int 27 d x = 27 x

= \frac{2}{5} x^{\frac{5}{2}} + \frac{9 x ^{2}}{2} + 18 x^{\frac{3}{2}} + 27 x

解答补常数

= \frac{2}{5} x^{\frac{5}{2}} + \frac{9 x ^{2}}{2} + 18 x^{\frac{3}{2}} + 27 x + C

x \to - 4 lim (x^{3} + 4 x^{2} - 8 x)

y ln (y) d x + (x - ln (y)) d y = 0

\frac{\partial}{\partial x} (x^{3} y + e^{x y^{2}})

\frac{d y}{d x} = 1 - 2 y

\int tan^{5} (8 x) sec^{4} (8 x) d x