integral of (x^3+3)^{1/4}*x^5

解答

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

\frac{1}{3} (\frac{4}{9} (x^{3} + 3)^{\frac{9}{4}} - \frac{12}{5} (x^{3} + 3)^{\frac{5}{4}}) + C

求解步骤

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

使用换元积分法

= \int \frac{( u - 3 ) u ^{\frac{1}{4}}}{3} d u

提出常数:

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

= \frac{1}{3} \cdot \int (u - 3) u^{\frac{1}{4}} d u

乘开

(u - 3) u^{\frac{1}{4}} : u^{\frac{5}{4}} - 3 u^{\frac{1}{4}}

= \frac{1}{3} \cdot \int u^{\frac{5}{4}} - 3 u^{\frac{1}{4}} d u

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

= \frac{1}{3} (\int u^{\frac{5}{4}} d u - \int 3 u^{\frac{1}{4}} d u)

\int u^{\frac{5}{4}} d u = \frac{4}{9} u^{\frac{9}{4}}

\int 3 u^{\frac{1}{4}} d u = \frac{12}{5} u^{\frac{5}{4}}

= \frac{1}{3} (\frac{4}{9} u^{\frac{9}{4}} - \frac{12}{5} u^{\frac{5}{4}})

u = x^{3} + 3

代回

= \frac{1}{3} (\frac{4}{9} (x^{3} + 3)^{\frac{9}{4}} - \frac{12}{5} (x^{3} + 3)^{\frac{5}{4}})

解答补常数

= \frac{1}{3} (\frac{4}{9} (x^{3} + 3)^{\frac{9}{4}} - \frac{12}{5} (x^{3} + 3)^{\frac{5}{4}}) + C

l a pl a ce t r an s f or m t e^{- 3 t} cos (2 t)

\int_{1}^{\infty} \frac{1}{4 ^{x}} d x

\int 3 - 3 x d x

\int x cos (2 x) d x

d er i v a t i v e y = e^{x^{6}}