integral of (x-3)sqrt(2x+1)

解答

\int (x - 3) 2 x + 1 d x

\frac{1}{5} x (2 x + 1)^{\frac{3}{2}} - \frac{16}{15} (2 x + 1)^{\frac{3}{2}} + C

求解步骤

\int (x - 3) 2 x + 1 d x

乘开

(x - 3) 2 x + 1 : x 2 x + 1 - 3 2 x + 1

= \int x 2 x + 1 - 3 2 x + 1 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 2 x + 1 d x - \int 3 2 x + 1 d x

\int x 2 x + 1 d x = \frac{1}{15} (3 x - 1) (2 x + 1)^{\frac{3}{2}}

\int 3 2 x + 1 d x = (2 x + 1)^{\frac{3}{2}}

= \frac{1}{15} (3 x - 1) (2 x + 1)^{\frac{3}{2}} - (2 x + 1)^{\frac{3}{2}}

化简

\frac{1}{15} (3 x - 1) (2 x + 1)^{\frac{3}{2}} - (2 x + 1)^{\frac{3}{2}} : \frac{1}{5} x (2 x + 1)^{\frac{3}{2}} - \frac{16}{15} (2 x + 1)^{\frac{3}{2}}

= \frac{1}{5} x (2 x + 1)^{\frac{3}{2}} - \frac{16}{15} (2 x + 1)^{\frac{3}{2}}

解答补常数

= \frac{1}{5} x (2 x + 1)^{\frac{3}{2}} - \frac{16}{15} (2 x + 1)^{\frac{3}{2}} + C

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