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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

使用换元积分法

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

使用换元积分法

= \int (v + 1) v^{\frac{1}{4}} d v

乘开

(v + 1) v^{\frac{1}{4}} : v^{\frac{5}{4}} + v^{\frac{1}{4}}

= \int v^{\frac{5}{4}} + v^{\frac{1}{4}} d v

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 v^{\frac{5}{4}} d v + \int v^{\frac{1}{4}} d v

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

\int v^{\frac{1}{4}} d v = \frac{4}{5} v^{\frac{5}{4}}

= \frac{4}{9} v^{\frac{9}{4}} + \frac{4}{5} v^{\frac{5}{4}}

代回

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

化简

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

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

解答补常数

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

作图

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